Green’s Dictionary of Slang

(US) a law enforcement officer of the US Treasury Department; also attrib.

1935		Hartford Courant (CT) 8 Dec. sect 6 1/1: The T-men (Treasury Men) were organized [in 1934] and the drive on bootlegging opened.
1948		Annuities for Investigatory Personnel of Treasury Dept. 24: I was in New York on a field trip recently, in fact the week before last. I ran into a T-man.
1950		CBS Radio in Terrace Radio Program Openings and Closings, 1931-1972 (2003) 236: Announcer (Bob Lemond) T-Man, starring Dennis O’Keefe. T-Man, the law enforcement agent of the Treasury Department, skilled fighter against crime, relentless enemy of the underworld.
1952		(con. 1920s) ‘Harry Grey’ Hoods (1953) 303: A batch of dough that the ‘T’ men themselves won’t be able to tell the difference.
1958		(ref. to 1931) A. Tully Treasury Agent 93: ‘Don’t bother me,’ Waxey [Gordon] told a T-man who came to call. ‘I don't keep any records’.
1962		P. Crump Burn, Killer, Burn! 344: Playing T-man guessing games.
1970		Field & Stream Mar. 10: Well, sir, the Treasury's anti-gun film is pure melodrama [...] The Congressman [...] demanded that the T-Man tell all present what the intent of the movie.