Maybe he had presence of mind to have brought hats with him to the cemetery knowing he was going to leave the old hat behind on the casket. One on his head, one in hand

NoI don't remember that. But I still don't understand why, when all the three surviving members of the "squad" put their hats on the casket, Max then puts on his hat and does the last scene with his wifewith the hat on. He should have been hatlesslike the rest of his squad.
Ciao, e buon auguri

They talk about that in the beginning of the movie. All the guys enjoy walking around buck-naked down at the precint while hanging their hats on their hard man meat. The detective who dies at the end tells the others "Man gosh oh golly if I ever die I want these sacred hats we wear on our willies to sit on thy grave, hand to God." All the other guys break out in laugter at this but after a while Nick Nolte's character quiets everybody down and says in a low nd raspy voice "We will my friend, cin'-mon' buns." Then they got dressed and went out for steaks and cigars. Don't you remember that?????????

Who cares, it's a movie.

I just took it as some ritual, between the four of them, made up on the spot.
Life is never fair, and perhaps it is a good thing for most of us that it is not.