The 26-year-old Allard pitched in just four games last season for Atlanta.
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He spent four seasons with Texas (2019-22) and is 9-24 with a 6.10 ERA and one save over parts of six major league seasons with the Braves and Rangers.
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The 6-foot-1, Anaheim, Calif., native was originally selected by the Braves in the first round (No. 14 overall) of the 2015 MLB Draft.
The Associated Press contributed to this report.