Wilmet learned a slider from his brother, Steve, who played in the Los Angeles Dodgers system. Wilmet attended West De Pere High School in Wisconsin and played college baseball at Des Moines Area Community College in Boone, Iowa.
Having been undrafted out of college, Wilmet toured as a musician for a few years before focusing on playing amateur baseball in Green Bay, Wisconsin. On the strength of his amateur performance, he was offered contracts by the Dodgers and the New York Mets. After flipping a coin, he signed with the Mets. When the Mets released him, he drove to St. Louis Cardinals spring training and camped out in the back of his Ford E-Series van until he was offered a contract.
In his Major League debut on July 25, 1989, Wilmet helped set a Texas Rangers franchise record for strikeouts in a nine-inning game. After Nolan Ryan recorded 14 strikeouts in six innings and Kenny Rogers added three, Wilmet added the team's eighteenth strikeout in the ninth inning.

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