Green’s Dictionary of Slang

(US) a hat, esp. a soft, felt one.

1883		Bristol Magpie 1 Mar. 13/1: [A]a stout short-sighted fiend in human form [sat] dead on the cherised pericranium coverer. Alas! poor ‘tile.’ Tableau! Ten thousand apologies; and a ‘crusher’ .
1916		J. Lait ‘Omaha Slim’ in Beef, Iron and Wine (1917) 112: [He] was distinguished among the hobos because he carried a pocket comb, and, from time to time, would take off his worn and shiny crusher and caressingly comb his shock.
1947		M.H. Boulware Jive and Sl.