More literally "and having put on the new" (self is not in the Greek but is added in the NAS).

This verse shifts from the negative (laid aside) to the positive (put on).  With the stripping off of the old nature there has come a new nature, the new man.  We have laid aside the old garment, the "hand me down" rags from Adam and have put on the new garment, the new man in Christ.

Have put on  (1746) (enduo from en = in + dúo = to sink, go in or under, to put on) means literally to clothe or dress someone and to put on as a garment, to cause to get into a garment (eg, Lu 15:22 where the father says "quickly bring out the best robe and put it on him...").

In the middle voice (as here in Col 3:10) it means to clothe oneself with something, in this case the "new man", who is clothed in the robes of Christ's righteousness and now needs to practice what this privileged position entails, i.e., to manifest His righteousness each day toward God and toward men in everyday life.

Aorist tense indicates this putting on is a past completed action and includes the idea that this action was decisive.

When did we put on the new? This "putting on" occurred the moment the old self (old man) died with Christ, at the time of regeneration by grace through faith. In verse 9 above, the "laying aside" is also aorist tense, and one can deduce that the action of both verbs took place at the same time in the past, corresponding to the moment we believed in Christ.

S. Lewis Johnson explains enduo as used in Colossians 3:10 writing that...

The believer, having been severed from his connection with Adam the first, has now been clothed with and joined to Adam the last in all His fragrance and beauty. (Bibliotheca Sacra: volume 121, issue 481, 1964).

And so what is true about every believer is that they have put on the new self, which describes their new position as children of God, His new creations. We now are secure in an inseparable union with Christ and are fully identified with His life. Our old self has been laid aside. Because of our new position, we now are in possession of divine, supernatural power which gives every believer the potential to walk in newness of life (see note Romans 6:4).

The NLT says it this way

"In...place (of the old, evil nature) you have clothed yourselves (this happens at a moment in time) with a brand-new nature that is continually being renewed (this describes the ongoing process known as sanctification) as you learn more and more about Christ, Who created this new nature within you."

THE ARMOR OF LIGHT

Romans 13:12 The night is almost gone, and the day is at hand. Let us therefore lay aside (cast off, drop, fling away, renounce) the deeds of darkness (all the filthy garments of worldliness—that is, everything associated with unrighteousness and evil -- in the context of Col 3:10 this would include lying) and put on the armor of light. (See notes)

CHRIST HIMSELF AS OUR GARMENT

Galatians 3:27 For all of you who were baptized into Christ have clothed yourselves with Christ. (Comment: This putting on refers to salvation, at which time the Spirit replaced our filthy rags of sin with the righteousness of Christ - this is now and forever our new position before God. He sees us in Christ's righteousness - the theologians refer to this as positional truth = past tense salvation = justification).

Romans 13:14 But put on (our practice = present tense salvation = progressive sanctification - put Him on each morning and every moment of the day - aorist imperative [middle voice = you initiate the action and participate in the result = put Him yourself] ) the Lord Jesus Christ, and make no provision for the flesh in regard to its lusts. (See notes)

THE NEW SELF

Ephesians 4:24 and put on (not a command - aorist tense) the new self, which in the likeness of God has been created in righteousness and holiness of the truth. (Comment: As discussed in the notes there is debate between excellent commentators, some favoring this putting on as indicative of positional truth and others favoring it as calling for this to be our practice - progressive sanctification or present tense salvation).

Colossians 3:10 and have put on (past tense salvation = positional sanctification = our position now and forever in Christ - see our practice in Col 3:12) the new self who is being renewed to a true knowledge according to the image of the One Who created him (See notes)

Colossians 3:12  And so, as those who have been chosen of God (cf notes Eph 1:5), holy and beloved, put on (present tense salvation  = progressive sanctification = our practice - a command be clothed [middle voice = clothe yourself] now = aorist imperative) a heart of compassion, kindness, humility, gentleness and patience (See notes)

THE BREASTPLATE OF FAITH AND LOVE

1Thessalonians 5:8 (note) But since we are of the day, let us be sober, having put on (at the time of our new birth = justification = our position = past tense salvation) the breastplate of faith and love, and as a helmet, the hope of salvation.

IMPERISHABLE, IMMORTAL GARMENTS

1 Corinthians 15:53 For this perishable must put on (glorification = future tense salvation) the imperishable, and this mortal must put on immortality. 54 But when this perishable will have put on the imperishable, and this mortal will have put on immortality, then will come about the saying that is written, "Death is swallowed up in victory.

What a "wardrobe" God has made available for believers! We're the "best dressed" folks in the world and most of us don't even know it! And the best is yet to come for John writes...

GARMENTS OF FINE LINEN, WHITE AND CLEAN...

Revelation 19:14 (note) And the armies (this is us, those redeemed by the blood of the Lamb) which are in heaven, clothed (enduo) in fine linen, white and clean, were following Him (the Lamb = Faithful and True = the Word of God) on white horses. (Comment: This incredible historical event will occur at the end of the 7 year period, Daniel's Seventieth Week, and marks the defeat of the antichrist and his armies and the inception of Messiah's Millennial Reign)

The garments believers are now wearing (figuratively, spiritually) are a picture of every believer's vital mystical spiritual union with Christ which began at the time of regeneration (the new birth). All believers have been irrevocably, intimately united with Christ at the moment of salvation (past tense salvation = positional sanctification = justification = the new birth). Every believer now stands (our position) before God clothed with Christ's righteousness, complete in Christ.

Thus in Romans 13 Paul writes that

The night (of man's depravity, of this present evil age) is almost gone, and the day (of Christ's return and reign) is at hand. Let us therefore lay aside (cast off, drop, fling away, renounce) the deeds of darkness  and put on (enduo) the armor of light (which equates with the protection that practical righteousness and holy living imparts) (see note on Romans 13:12)

 Paul adds that they are to

Put on (enduo - aorist imperative - do it now) the Lord Jesus Christ, and make no provision (forethought, thought in advance) for the flesh in regard to its lusts ("don’t think of ways to indulge your evil desires", NLT)." (see note Romans 13 14)

In short, if we are Christians we have already put on the Lord Jesus Christ as stated in (Gal 3:27) Just as a garment which one puts on envelops the person wearing it and defines his appearance, so the person baptized in Christ is entirely taken up in Christ and in the salvation brought by Him. But this passage in Romans has reference to practical day-to-day, repeated putting on of Christ.

Ray Stedman gives the following illustration:

"When I get up in the morning I put on my clothes, intending them to be part of me all day, to go where I go and do what I do. They cover me and make me presentable to others. That is the purpose of clothes. In the same way, the apostle is saying to us (see note Romans 13:12, 13; 14), “Put on Jesus Christ when you get up in the morning. Make Him a part of your life that day. Intend that He go with you everywhere you go, and that He act through you in everything you do. Call upon His resources. Live your life IN CHRIST.” (Stedman, Ray C. From Guilt to Glory. Vol 2. p136. Waco, TX: Word, 1978) (Bolding added)

Kent Hughes interprets the putting on of the new self in Ephesians 4:24 (see note) as follows...

"The fact is, we have this new self if we are Christians. We received the old man at birth, and we were given the new man in our heavenly birth. The new man is not our work — it is God’s creation and gift. Our task is not to weave it, but to wear it. Paul is commanding a daily appropriation of that which we already possess...We have our part to do in dressing ourselves with the divine wardrobe, for here clothes do make the man — and the woman! We must daily set aside the rotting garments of the old man. We must formally reject sensuality and selfish pride and materialism and bitterness. We must read the Word and ask God to to renew our minds through the Spirit. We must work out our salvation by doing those things that will develop a Biblical mind. We must put on our new, shining garments of light. We must put on what we are!" (Hughes, R. K.: Ephesians: The Mystery of the Body of Christ. Crossway Books or Logos)

Because of their privileged position "as those who have been chosen of God, holy and beloved" Paul calls on the Colossian saints to put this truth into practice and to

put on (enduo - aorist imperative - do it now) a heart of compassion, kindness, humility, gentleness and patience... (see note Colossians 3:12)

In Colossians 3:10 (note) Paul explained that believers have put on the new man and here in Col 3:12ff he shows them practically how this new man is to live.

Thayer says that put on here in (Colossians 3:12 [note]) means

to become so possessed of the mind of Christ as in thought, feeling, and action to resemble Him and, as it were, reproduce the life He lived.

And truly, how else is it possible to live out the "put on's" which reflect the lifestyle of the "new self"?

To review this section, the verbs translated put off and put on  both are aorist tense and indicate a completed past event. When any sinner trusts Christ, at that moment they put off the old self and put on the new. At that moment, the old self has been rendered powerless and the new self is now to be in control (read especially Romans 6 - see notes Romans 6). In Christ the believer has been set free from the dominion and power of "sin". Because of our permanent identification and union with Christ, we now have the power of His Spirit and the responsibility to conduct ourselves as a new man in Christ (this is called sanctification and is alluded to in the next part of this verse). We are not told to feel that these things are true or even to fully understand them. But we are told to live them out by grace through faith. That is what Colossians 3 is all about -- walking the talk. Living out the life that is in us - Christ is us the hope of glory. As we walk in a manner worthy we will grow in the knowledge of God (see note on Colossians 1:10) and in so doing we will come to understand more and more the depth of the truth that we are now in Christ. (see also in Christ and in Christ Jesus)

New (3501) (neos) signifies new in respect to time (contrast kainos = new in respect to quality) and describes that which has recently come into existence but for a relatively short time (eg, "new wine" Mt 9:17).

Neos is used 24 times in NT (1x Mt; 1x Mk; 6x Lu; 1x Jn; 1x Acts; 1x 1Cor; 1x Col; 4x 1Ti; 2x Titus; 1x Heb; 1x 1Pe)

Neos can also refer to one who is in the early stages of life (i.e., young) and this use accounts for many of the NT occurrences.

"New [self]" ("self" is added by translators) describes who believers are by virtue of their union with Christ.

The new self continually being renewed describes the process of sanctification. (Click here to compare the "three tenses of salvation")

Vine says that the use of neos

stresses...the fact of the believer’s new experience, recently begun and still proceeding. (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson or Logos)

MacDonald explains that

Just as the old man refers to all that we were as sons of Adam, with an unregenerate nature, so the new man refers to our new position as children of God. There has been a new creation, and we are new creatures. God’s purpose is that this new man should always be growing more and more like the Lord Jesus Christ (Ed note: Practical sanctificaition). We should never be satisfied with our present attainments, but should always press on to the goal of increasing conformity to the Savior. He is our example and the rule of our lives. In a coming day, when we stand before the Judgment Seat of Christ, we will be judged not by how much better our lives were than others but rather by how our life measured up to the life of the Lord Jesus Himself. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson or Logos)

MacDonald goes on to quote the "Daily Notes of the Scripture Union" which records that

The image of God is not seen in the shape of our bodies, but in the beauty of the renewed mind and heart. Holiness, love, humility, meekness, kindness, and forgiveness—these make up the divine character.