Skinner spent nine seasons on the Indians' Major League staff, with stints as the third-base coach, interim manager and bench coach.
Skinner appeared in 564 games as a catcher with the White Sox (1983-86), Yankees (1986-88) and Indians (1989-91).
Also on Monday, the White Sox announced that Brandon Moore will be the hitting coach at Double-A Birmingham. Moore spent the last seven seasons in the Pirates' Minor League system.
Moore returns to the White Sox organization where he was the hitting coach for Rookie League Bristol (2001-02) and Class A Kannapolis (2003-04).
Other changes Monday included the White Sox naming Tim Laker as their Minor League roving hitting instructor. Andy Tomberlin will be on Skinner's staff as the hitting coach and former White Sox closer Bobby Thigpen will be the pitching coach at Birmingham.
Buddy Bell, who spent 18 seasons in the Major Leagues as a player and nine more as a manager, returns for his fourth full season as the club's director of player development.