Billycock

Bil"ly*cock, n., or Bil"ly*cock hat`. [Perh. from bully + cock; that is, cocked like the hats of the bullies.]

Defn: A round, low-crowned felt hat; a wideawake. "The undignified billycocks and pantaloons of the West." B. H. Chamberlain.

Little acquiesced, and Ransome disguised him in a beard, and a loose set of clothes, and a billicock hat. Charles Reade.