Thanks for the help, all.

I tried the coat hangar trick and got a fair sized wad of hair, but it
still drained slowly.

I went to Menard's and got a cheap 15' snake. It might have been about
5' in when I retrieved what initially looked like a very large rat,
but was actually a gigantic loathsome mass of hair and random sewer
crud.

Shower drains great now.

Thanks for the tips and hints.