Of the group, all but Maile have been with the team this season.
Maile, 24, stands 6-foot-3, 226 pounds and hit .207 with five home runs and 29 RBIs in 89 games for Triple-A Durham this season.
Mahtook, 25, hit .189 with three home runs and six RBIs in 14 games for the Rays this season. At Durham, he hit .249 with four home runs, 45 RBIs and 10 stolen bases.
Moore, 26, returned to the Rays from Tommy John surgery on July 2 to make the first of six starts with the club, in which he went 1-3 with an 8.