Green’s Dictionary of Slang

1. (Aus./US) an advertising leaflet, a flyer.

1879		Bernalillo Mirror 8 Mar. 3/1: Circulars, statements, dodgers, tickets, and all kinds of printing done at the Mirror office in the best style and at reasonable prices [DA].
1883		Nat. Police Gaz. 20 Oct. 7/4: Perched on the fire escape a couple of pretty and stylishly dressed girls scattered advertising dodgers of various businesses.
1889		Music Hall & Theatre Rev. 2 Mar. 38/2: [I]n Sacarmento he got hold of a ‘dodger’ — that is, a small handbill.
1900		Bulletin (Sydney) 17 Mar. 32/3: Jarvis, you will see the printer and order three-thousand more dodgers.
1910		Sun. Times (Perth) 22 May 2nd sect. 12/5: Miss Matters is aa intelligent young Australian, and her enthusiasm over [...] distributing ‘dodgers’ from a balloon is a matter for mild wonder.
1912		in W.C. Fields By Himself (1974) 46: Passing circulars [...] I went to a pond where a number of young people were skating and threw an armful of dodgers on the ice.
1933		H. Asbury Barbary Coast (2002) 127: A typical Tetlow dodger, issued in 1862, described the Bella Union’s theatrical fare.
1948		Oklahoma City Times 14 June 13/8: The scandal sheet, usually a one-page dodger, devotes a great deal of black ink to besmirching a candidate’s character [DA].
1961		F.J. Hardy Hard Way 107: ‘Give us a few of those dodgers.’ [...] I handed him a bundle of leaflets.
1965		L. Haylen Big Red 185: He held up a tattered dodger which said: ‘Join the I.W.W. Only fools and horses work hard!’.

2. (US Und.) a ‘wanted’ flyer distributed to law enforcement agencies, post offices etc.

1933		Ersine Und. and Prison Sl.
1947		F. Brown Fabulous Clipjoint (1949) 159: Yes, but – I’d be wanted [...] There’d be dodgers out for me.
1950		Goldin et al. DAUL 60/1: Dodger. A handbill for a wanted criminal.
1960		H. Whittington Night for Screaming (2006) 56: They would check the files at headquarters. They would find the wanted dodger on me.