Frazier signed a contract with the Charleston Dirty Birds of the Atlantic League on Thursday.
Frazier elected to become a free agent Monday.
The White Sox sent Frazier outright to Triple-A Charlotte on Friday.
Frazier went 1-for-3 with a steal in Tuesday's win over the Guardians.
Frazier is starting in right field and batting eighth for the White Sox on Tuesday versus the Guardians.
Frazier was called up from Triple-A Charlotte on Sunday, LaMond Pope of the Chicago Tribune reports.
Frazier signed a minor-league contract with the White Sox on Friday, James Fegan of The Athletic reports.
Frazier was released from his minor-league contract by the Rangers on Monday, Kennedi Landry of MLB.com reports.
Frazier has been told he will not make the Rangers' Opening Day roster, Evan Grant of The Dallas Morning News reports.
Frazier is using "Clint" as his first name this year after switching to Jackson (his middle name) in June of last year, Levi Weaver of The Athletic reports.