Forgot to mention this in my trip report, but we found that there is actually a flip drawer in every FJ seat where you can put stuff. On the back on the seat (think, where the small of your back would be when sitting). It flips out, and you can easily store a hat or purse in there.
I've been on FJ 20 or 30 times and never knew this. But one of these last times we spent more time than usual in the Gryffindor common room, and one of the paintings mentions it (they also say you can store your wand in there!).
I wonder if they hoped for this to be the storage device before they figured out they needed lockers. Nonetheless, they still work!