In a three-season career, Minchey posted a 3–7 record with 38 strikeouts and a 6.75 ERA in 15 appearances, including 12 starts, one complete game, and 64.0 innings of work. Following his majors career, Minchey played in Japan for seven seasons. He had a 74–70 record with 626 strikeouts and a 3.64 ERA in 1213⅓ innings, collecting two 15-win seasons and leading the Japanese Pacific League with a 3.26 ERA in 2001.
Minchey also pitched in the Montreal, Atlanta, Boston, St. Louis and Colorado minor league systems from 1987 to 1997. He went 112–104 with 1141 strikeouts and a 3.54 ERA in 1832⅔ innings, and posted identical records of 15–6 in 1992 and 1997. Overall, Minchey collected 189 wins in a span of 16 years. Since 2007, he has worked as a scouting supervisor for the Cleveland Indians in Japanese baseball.