Clint, I'm thinking it may have been a seat protector for the top section where your head might rest. Possibly to keep the seat cleaner where your head would be. It's what came to mind for me.
Jon
If that is what it is, then it is an antimacassar.
If you are wondering "what is a macassar?"
Macassar oil is a compounded oil used primarily by men in Victorian and Edwardian times as a hair conditioner to groom and style the hair.
Probably more than you wanted to know.