Winkler exercised the opt-out in his minor-league contract with the Rangers on Wednesday and has become a free agent, Levi Weaver of The Athletic reports.
The 32-year-old joined Texas on a minor-league deal in mid-March but was unable to earn a chance in the majors. Winkler had a 3.50 ERA, 1.39 WHIP and 26:13 K:BB over 18 innings for Triple-A Round Rock. He'll now look to latch on with another organization.