I'm not sure if this was a game used hat or not, but it sounds similar to a situation I just had. I got the July 4th game used hats of two local minor leaguers with potential in a charity auction. They were having an aftergame signing that weekend, so I took the hats to get signed. As you can see, I chose to go under the brim and had them mark it as a Game Hat.
Here is what the hats looked like new, before being worn on the July 4th game:
And here is how I had them signed (sorry the colors are off, phone pic):
