I would think hooks on the wall would be the easiest idea. If you don't want to put holes in the wall, try Command hooks.
Use a long thick ribbon and clip on hats with close pins and then hang the ribbon up.
Coffee mug rack. The rack says “mugs on mugs” because I am always putting friends pics on mugs. These are my quickie our-the-door erranding caps. Have waaaay more hats and am crafting a new hat rack right now in an unconventional way with an upside down shelf from Hobby Lobby and clothes pins for the nicer non-ballcap hats with wide brims.