Green’s Dictionary of Slang

(orig. Aus.) a hat; a slouch hat.

1882		‘The Jargon of Thieves’ in Derry Jrnl 8 Sept. 6/6: A soft hat is a ‘caddie,’ and a stiff hat is a ‘dicer’ .
1893		P.H. Emerson Signor Lippo 55: One day he walked straight into this kitchen clobbered in a pair of rounds, tight to his legs, a black frock isleim [...] and a long sleeve cadi on his napper.
1910		F. Murray [perf. Harry Champion] ‘Ginger you're balmy’ 🎵 I've been walking about the streets without a hat / I do without a caddy, and it saves me half a quid.
1949		Monteleone Criminal Sl. (rev. edn).
1977		D. Powis Signs of Crime 176: Caddy A hat.