Green’s Dictionary of Slang

dicer n.1

also dice
[its resemblance to a dice-box]

a hat.

[US]St Louis Globe-Democrat 19 Jan. n.p.: Whereupon, the party addressed takes an inventory of his friend’s clothes and [...] as to his head-covering, he thinks had better ‘shoot that tile’ and buy a new ‘dice,’ as ‘cadies’ of that style are out of fashion.
[UK]Puck XV-XVI 68/1: And he girdeth up his loins and he putteth on his dicer, even his dicer of silk.
[US]Lantern (N.O.) 9 July 2: Now Lehman wears his dicer down over his eyes.
[US]Ade Artie (1963) 56: Tommy—new suit, new white necktie, new dicer, new shoes.
[US]E.W. Townsend Sure 147: Whiskers holds out his hat about five feet up. Say, she let go, and de toe of her shoe runs tru de top of de silk dicer.
[US]Van Loan ‘On Account of a Lady’ in Taking the Count 124: He tipped his mitt de minute he took off his dicer.
[US]P. & T. Casey Gay-cat 302: Dicer—a hat Synonymous with ‘lid,’ or ‘beany’.
[US]A.J. Pollock Und. Speaks 30/2: Dicer, a derby hat.
[US]C.S. Montanye ‘Shoulder Straps’ in Thrilling Detective Feb. 🌐 McClain [...] hung his dicer on the chromium hat-rack.
[US]Monteleone Criminal Sl. (rev. edn).
[US]A.J. Liebling ‘Days with the Daydaybay’ in Just Enough Liebling (2004) 187: Of course, the dicer fell to the floor, where it rolled between a woman’s feet.
[US]Ragen & Finston World’s Toughest Prison 796: dicer – A hat.