Manco, a.k.a. the Man With No Name, wears this for pretty much the entirety of For a Few Dollars More. The one exception I could find was when he was snooping around after dark in Agua Caliente, and this seems to have significance because when El Indio catches him, he hands Manco this object and orders him to put it back on.
I have no idea why El Indio told him to do that. Does the fact that it fits prove something about what Manco was up to? Does wearing it make him better at shooting or fist-fighting somehow? Or was it just a piece of small talk, like "by the way, you forgot this"?
Here's a slightly clearer shot of it from Manco's first scene in the movie:
The other reason I'm curious about this is that I'm fairly sure he was not wearing this in A Fistful of Dollars or The Good, The Bad and The Ugly, and I believe his right hand was injured in A Fistful of Dollars after the Rojos captured him, so it might even be a tiny piece of evidence that the Man With No Name really is the same character in the two "Dollars" movies.