The team announced the move on Thursday. Cobb had been on the DL since April 13.
To make room on the roster for Cobb, outfielder Kevin Kiermaier was optioned to Triple-A Durham after Wednesday night's game against Oakland.
Hellickson threw 34 pitches during his first batting practice session Thursday. The right-hander, who expects to return before the All-Star break, had arthroscopic right elbow surgery in January.
Moore underwent left elbow ligament replacement surgery in April and could be sidelined 12 to 15 months.