However, he did not stay at Medford long, as after stops in Modesto and Huntsville, he made his final start of his first season in professional baseball pitching for the Triple-A Tacoma Tigers and finished the year a combined 8–2 with a 2.84 ERA and 47 strikeouts in 85+2⁄3 innings.

[5] In 1987, Tapani began at Single-A Modesto, despite dominating at the level the previous year, and turned in another good season with a 10–7 record, a 3.76 ERA, and 121 strikeouts in 148+1⁄3 innings.

He entered the game in the first inning after starter Bob Ojeda had given up eight runs to the Houston Astros while only getting two outs.

Cy Young Award winner Frank Viola from the Minnesota Twins to the Mets in exchange for major league swingman Rick Aguilera, minor league relievers Tim Drummond and Jack Savage, and young starter David West.

After his promising debut, Tapani was made a fixture of the rotation in 1990 and finished the season 12–8 with 4.07 ERA – good enough for fifth place in the AL Rookie of the Year balloting, a distant 131 votes behind the winner, Cleveland Indians' catcher Sandy Alomar Jr.

After again being granted free agency following the 1996 season, Tapani moved across town and signed a five-year contract with the Cubs on December 13.

In 1991, with the Twins, Tapani had his lowest ERA of 2.99 to go with a 16–9 record and seventh place in the 1991 Cy Young Award voting.

After retirement, Tapani returned to his home in Minnetonka, Minnesota, to live with his wife, Sharon, and three children, Sarah (b 1991), Ryan (b 1994), and Luke (b 1996).

Ryan spent 3 years pitching in the Nationals minor league farm system, along with one for the Kane County Cougars of the American Association of Professional Baseball.