In several letters (Gal 3:27, Rom 13:14, Eph 4:24, Col 3:10) Paul refers to "putting on Christ" or "putting on the new self". Putting on Christ sounds like putting on a coat and, in fact, that can give us a good idea of what Paul meant.
First, putting something on is a conscious act. I cannot sit around waiting for my coat to come to me. I cannot walk around my living room hoping that I'll accidentally stumble into my coat. I have to purposefully put it on.
A good winter coat is all-encompassing and protective. A coat is both your public "self" when you're outside - what the world can see of you - and your inner protection against the cold and buffeting wind outside.
Lastly, what you put on is a visible and integral part of you. When I walk down the street, no one says "there's Joe and his coat". I have put it on and everyone can tell it is a part of me. No one calls it out as a seperate thing because it is clearly part of myself.
So as Paul tells us, put on Christ.