I found some hat patterns online, thanks to intrepid link following...a sewing blog linked to a sewist's flickr set, and whilst perusing her photos, I came upon some hats that she had sewn. In her comments she provides some handy links to other hat patterns. I decided to give one a shot, despite instructions being written in Japanese, using some scrap fabric that I had on hand. Mickey is modeling the outcome below. It's reversible! And too big for my head. According to the pattern (I think), you were supposed to use six pieces. Six pieces created a hat that was too small to fit around my head, so I expanded to eight pieces. It's now just a tad too large. It will work, but not on a windy beach day.
I will definitely make more hats! This one took absolutely no time at all, so it's very handy when you want a quick project.