Green’s Dictionary of Slang

bay n.

SE in slang uses

In phrases

half the bay over (adj.) [var. on half seas over adj. (1)]

drunk.

2004

‘Drunk Words’ at www.manwalksintoapub.com 🌐 Half the bay over.

over the bay (adj.) (also over the dam) [var. on half seas over adj. (1)]

(US) drunk, tipsy.

1833		A. Greene Life and Adventures of Dr Dodimus Duckworth II 176: He was seldom downright drunk; but was often [...] a little over the bay [...] half way over the dam.
1844		‘Jonathan Slick’ High Life in N.Y. I 167: Some of ’em were purty well over the bay.
a.1856		Burlington Sentinel in Hall (1856) 461: We give a list of a few of the various words and phrases which have been in use, at one time or another, to signify some stage of inebriation: Over the bay.
1891		Salina Dly Republican (KS) 25 Sept. 3/2: Over the bay — Drunken man on his way home, tacking from one side of the street to the other.
1915		M.G. Hayden ‘Terms Of Disparagement’ in DN IV:iii 220: over the bay, intoxicated.
1927		‘Sl. Expressions for Drunk’ in New Republic in AS XVI:1 (1941) 9 Mar. 70: [...] over the Bay.