Green’s Dictionary of Slang

well under way adj.

(US) drunk.

1851		Kalida Venture (OH) 11 Apr. 2/4: Drunk [...] well under way.
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: [...] well under way.