Green’s Dictionary of Slang

(US tramp) bay rum, a hair tonic; thus bay horse jockey, a drinker of bay rum.

1957		Social Problems V 314: Dehorns [...] are habitual drinkers of [...] bay rum (‘bay horse’) and canned heat. A man on bay rum is referred to as a ‘bay horse jockey’ [HDAS].
1965		S.E. Wallace Skid Row 203: Bay horse—bay rum. Bay horse jockey— one who drinks bay rum.