Adcock joined the team for Thursday's game against the New York Yankees.
The 26-year-old Adcock was 1-0 with one save and a 2.76 ERA in 13 games at Triple-A Round Rock.
Adcock pitched 36 games for Kansas City in 2011 and 2012, and signed with Texas in the offseason. He had a rib removed this spring in surgery for thoracic outlet syndrome, a nerve problem that was causing numbness in his right arm and hand.
Rangers manager Ron Washington says Adcock can be brought in to pitch multiple innings.
Texas traded Soria to the Tigers on Wednesday night for two young pitchers. To make room, first baseman Mitch Moreland was transferred to the 60-day disabled list. He had season-ending surgery on his left ankle last month.