I hate wearing hats, and baseball caps never fit right on my melon. But I go to WDW without any shame, wearing a big floppy wide-brimmed hat (like if Indiana Jones was a nerd) whenever I'm in the sun. If it's really hot, I buy one of those cheap straw cowboy hats -- they get more airflow and provide the necessary shade. I also carry a backpack and tuck away my embarrassing lid as soon as I'm out of direct sun.