The 6-foot-7 right-hander was impressive in the Minors this year, posting a 3.47 ERA with 66 strikeouts and 21 saves in 57 innings between Double-A New Britain and Triple-A Rochester.
Gardenhire said the key to Tonkin's success against the Rays was the use of his power slider, and he wants to see him using that pitch more moving forward.
"We were begging for sliders," Gardenhire said. "He kept throwing fastballs. But he finally did. And it's not even so much throwing it over for a strike, it's about showing them something different to set up that 95-mph fastball. If you keep pumping fastballs, they'll whack you. So he threw some sliders in the dirt and struck out [Evan] Longoria on one.
"So he has to keep spinning the ball, because there are a lot of guys who throw 95, but if you have a breaking ball, it gives you a chance to be one of those guys at the back of a bullpen."