This thing in the front hall coat closet has me stumped. It consists of a 2x4 piece of wood attached vertically to the wall, with a very thin piece of wood (almost veneer-thin) attached near the top of it. The attached piece extends higher than the 2x4, and the top of the 2x4 is cut into a rounded profile.

The only thing I can guess is that it is for hanging a hat or umbrella? I’ve added a dotted line to show where the wood ends behind the top piece.