Winkler signed a minor-league contract with the White Sox on Thursday and was assigned to Triple-A Charlotte.
Winkler has already made a pair of relief appearances for Charlotte, covering two innings while giving up two earned runs on two hits and a walk. Before catching on with the White Sox, Winkler compiled a 3.50 ERA and 1.39 WHIP across 18 innings at Triple-A Round Rock. He opted out of his minor-league contract with the Rangers on June 1.