Green’s Dictionary of Slang

(US prison) a smokeable cigarette end.

1950		Goldin et al. DAUL 45/1: Clincher. (P) A cigarette clenched between fingertips and extinguished to be smoked again. ‘I knew that phony big shot when he smoked clinchers (was penniless).’.
1955		‘Hal Ellson’ Rock 106: I light a clincher and go to the door.