Green’s Dictionary of Slang

bye n.

SE in slang uses

In compounds

bye-drink (n.) (also bye-drinking) [SE bye and/the bye]

an alcoholic drink, taken other than at mealtimes.

1751
18001850
1883
[UK]W. Kenrick Falstaff’s Wedding (1766) I iii: I could wish, nevertheless, old white wine stood higher in his lordship’s favour; that I may not be stinted at table, or in my by-drinkings.
[UK]Daily Tel. 10 Jan. 5 3: Our business men [...] take [...] many more ‘bye-drinks’ in the way of ‘sherry’ and ‘whiskey cold’ than is good for them [F&H].

In phrases