Kunkel is the son of Bill Kunkel, whose umpiring career overlapped the first season of Jeff’s major-league career. Jeff was the third pick in the nation in the June 1983 draft, but his offense was too weak for a regular job and he became a utilityman.
Given the opportunity to play more in 1989, Kunkel saw action in a career-high 108 games, splitting time with Scott Fletcher and Fred Manrique at short and appearing in thirty games in the outfield. He responded by batting .270 with eight homers in 108 games for the Rangers, but his playing time dipped in 1991 as his average fell below .200.