Green’s Dictionary of Slang

a soft felt hat with broad brim and low crown.

1838		W. Howitt Rural Life of England I 163: The farm-servant in his straw-hat, or his wide-awake.
1841		W.J. Neale Paul Periwinkle 41: His hat was like himself – wide awake.
1850		F.E. Smedley Frank Fairlegh (1878) 340: His head was adorned with one of those round felt hats, which exactly resemble a boiled apple pudding, and are known by the sobriquet of ‘wide-awakes,’ ‘cos they av’n’t got no nap about ’em.’.
1860		Cheshire Obs. 18 Aug. 8/1: Wide-awakes our head adorn / [...] / Feathers in our hats are worn, / Fast young ladies.
1864		J.R. Lowell Fireside Travels 117: He shook the water out of his wide-awake.
1865		R.S. Surtees Facey Romford’s Hounds 184: [He] crownded himself with a drab wide-awake, with an eagle’s feather in the parti-coloured band.
1865		‘Mr Carlisle’ Stray Leaves 19: Some wore a wide-awake, others a sort of pudding shape, quilted with cotton [...] and others, again, a sort of turban twisted round a common hat .
1876		M.E. Braddon Dead Men’s Shoes I 290: Is there so much difference between a chimney-pot hat and a wide-awake?
1880		R. Rowe Picked Up in the Streets 15: Did you notice [...] the man with the white wide-awake that was trying to pick a quarrel?
1889		Clarkson & Richardson Police! 321: A billycock ... A slouch, head-guard, wake.
1890		‘Rolf Boldrewood’ Colonial Reformer 102: You might [...] wear a wide-awake and a tourist suit at a flower-show.
1897		Truth (London) 10 June 35/1: There were many ‘clurks’ [...] extremely conscious of their [...] new pinched-in wideawakes.
1912		E. Pugh City Of The World 271: A nice old benevolent party in gold barnacles and a hat that is called a wideawake.
1917		‘Henry Handel Richardson’ Aus. Felix (1971) 209: He laid his bag down and hung up his wide-awake.
1920		Nebraska State Jrnl (Lincoln, NE) 6 June 18/4: When his fellow citizens wore bonnets with stingy brims [...] Father would [...] appear in a wideawake.
1936		(con. 1830s–60s) ‘Miles Franklin’ All That Swagger 55: He took to an outsize in straw wide-awakes that year and filled it with the cool leaves of the peppermint.
1938		X. Herbert Capricornia (1939) 117: There was Jock in the middle of the crowd, waving his wideawake and shouting.
1941		Western Mail (Perth) 7 Aug. 17s/1: While padding the hoof along the city streets [he] sought to conceal his doscomfiture by pulling the brim of his wideawake well down.