billycock hat

{ n., or . [ Perh. from bully + cock; that is, cocked like the hats of the bullies. ] A round, low-crowned felt hat; a wideawake. “The undignified billycocks and pantaloons of the West.” B. H. Chamberlain. [ 1913 Webster ]

Little acquiesced, and Ransome disguised him in a beard, and a loose set of clothes, and a billicock hat. Charles Reade. [ Webster 1913 Suppl. ]