It was between Conte and Coleman for me.
While I think that Conte is overall the better manager (based especially on his club career), Coleman did slightly better in this tournament, given the personnel at their disposal.
Conte had more options & players to choose from, even if they wouldn't have fit his tactics.
Both had a well working system and they stuck with it, but Coleman's plan B and tactical chances were just a bit better, although Italy showed against Spain that they can also play pretty well in a different way.