Green’s Dictionary of Slang

1. (US) an area of south-eastern NYC, known for its tanning industry.

1877

N.Y. Dly Herald 9 Sept. 6/3: Mr David G. Broome, confidential clerk of a large leather house in ‘the swamp’.

2. (US) a rough area of New Orleans comprising several blocks from the river.

c.1930

(ref. to late 19C) N. Kimball Amer. Madam (1981) 231: The Swamp was the favourite of the flatboat men, that and Gallatin Street, the toughest area ever in New Orleans [...] It was fighting with your teeth, handy-andy (blackjack), hogleg (pistol) or chiv (knife) your only friends in the Swamp.