This was <PERSON>'s vow on joining the night's watch
"Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands, father no children. I shall wear no crowns and win no glory. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the shield that guards the realms of men. 381
I pledge my life and honor to the Night's Watch, for this night and all the nights to come."
“shall not end until my death…”
When he died and came back from death, he had technically fulfilled his vow and was no longer bound to the Nights Watch. It was a clean slate for him. So, he did a
My watch has ended
Reasons as to why he didn't join the Nights Watch
1. He had more pressing things to deal with and now that he was given a new lease at life, and free from his vow, he is free to deal with the real threats such as his family, avenging the losses in his family, stability at Winterfell, the Night King and Wights beyond the wall among many other things that went int chaos while he was bound by oath to the Nights Watch. This is an opportunity for him to be at places where hes actually needed and not be in a place where he can be stabbed and killed.
2. He was betrayed and killed by his sworn brothers from the Nights Watch and he would be stupid and broken to join in the Nights Watch again. Also people would have to have guts to make <PERSON> stay clearly knowing he resurrected after being killed by them. He was called the "<PERSON>" but him dying was the last straw.
3. The vow says "i live and die at my post", he kind of died at his post.
So it only makes sense for <PERSON> to leave the Night's watch once he has a chance and not renew his vow after being killed by his own men 725