I decided to go with a wall-mounted hatrack, but to buy one with nice round supports (instead of hooks, which can poke a weird shape into your hat) cost more than I wanted to pay. So I rigged up my own. Here's how:
Regular old run-of-the mill cast iron garden hooks, yep, for hanging planters. I picked up five of these at a summer clearance for a couple bucks each. I hung them over my dresser, in a spot where my hats would be protected from exposure to direct sunlight and from the dry heat from the vent.
I traced around a latte bowl to create five circles of fabric. I measured five rectangles by wrapping my fabric around the bowl, marking where the ends met, and adding about an inch for seams. These sewed up in just a few minutes - no need to finish the bottom hems, just tuck them under about 1/2 - 1" in the next step. (fabric from IKEA)
This is a good use for leftover polyfill! Be sure and pack these suckers tight, or they won't keep their shape, and then what's the point?
I used embroidery thread to cinch the bottom - just a quick running stitch all around, with long ends left hanging. This will hold that folded edge under, giving you a nice finished edge on your hat rack.
Work that sucker over your hook, so that it's not going to shift, and then cinch tight and tie a knot with the long ends of the thread. It might be helpful to have another set of hands for this part.