Green’s Dictionary of Slang

lead sheet n.

[musical jargon lead sheet, a sheet of music containing the melodic line and lyric only; Mezzrow & Wolfe (1946) ‘As for lead sheet, it’s only one sheet of music out of a whole orchestration, with only the melody line on it, and hence thin enough to mean topcoat; whereas the full orchestration is, by contrast, thick and bulky, and could only mean a heavy overcoat’]

(US black) an overcoat or other outer garment.

[US]Cab Calloway New Hepsters Dict. in Calloway (1976) 257: lead sheet (n.): a topcoat.
[US] ‘Jiver’s Bible’ in D. Burley Orig. Hbk of Harlem Jive.
[US]Mezzrow & Wolfe Really the Blues 217: Me with this lead sheet on, tryin’ to scuffle up those two’s and fews for uncle so’s I can bail out my full orchestration.