MINNEAPOLIS -- As Trevor May was warming up in the bullpen before the ninth inning on Tuesday night, he felt that it was one of the more difficult situations this season to gauge how "hot" he would need to get -- that is, whether he'd actually enter the game or not. He ultimately did pitch in a 10-run game, a rather unfamiliar position for some late-inning relievers.

But May said that he almost treated the situation as more of a challenge in his mind, given the difficulties in his warmup routine and because he was feeling somewhat tired and sluggish that late in the game. And he knew that it would be a good chance for him to reinforce his strike-throwing mentality.

"It was a good opportunity to battle through that," May said. "I almost made it a higher-pressure situation for myself in order to test that."

May needed 13 pitches in a perfect ninth inning. No sweat.

"You almost end up in that kind of odd space where you're not really sure how to do your job, but he went out there and did it well," manager Rocco Baldelli said. "He went out there and threw strikes and made all of his pitches and did a nice job."

It marked another promising outing for the 29-year-old right-hander, who has been working to address his difficulties with falling behind hitters and facing elevated pitch counts all season due to walks and lengthy at-bats. He says that much of it is mental work to complement the progress he has made with his mechanics throughout the season to get his fastball more consistently up to 96-97 mph.

"I'm working very, very hard on mindset and really, really, really working on filling up the zone and making really quality pitches, and when I miss spots, making them just be slight misses, and if they're balls, they're where I'm not going to be hurt," May said. "I want to give myself that reinforcement every single time I throw the ball."

May looks to have turned a corner. He allowed just one hit in his past five outings, a span in which he has seven strikeouts and one walk in 6 2/3 innings. He has dominated the month of August, when hitters have combined for a .094 average against him with only three hits and three walks in 10 innings. He tossed a scoreless frame in Wednesday's 4-0 loss to the White Sox.

May has been near the top of the Major League leaderboards for lowest exit velocity allowed and lowest hard-hit rate allowed for much of the season, and he's continuing to internalize the fact that hitters have struggled to square him up when he's around the zone and that he doesn't need to fine-tune his pitch sequencing to strike everyone out.

That work is starting to show on the field.

"I wanted more strikeouts [at the start of the season]," May said. "But they don't matter. None of that stuff matters. What matters is getting off the field as quickly as humanly possible. So throwing over the plate hasn't even really hurt me that much. There's been a couple of misses that were crushed here and there, but that's baseball."

"Maybe it's not a rally squirrel."

Target Field's newest rodent resident made a cameo on the field for the second straight game on Tuesday night, when the so-called "rally squirrel" returned to the field and ran straight through the legs of Max Kepler at first base and into the White Sox dugout -- just before the Twins punched across four unearned runs to take the lead.

"We might end up with a little mascot, or I wouldn’t be surprised if you found some T-shirts with some players’ faces on it or someone doing something like that pretty soon," Baldelli said. "I wouldn’t mind if the squirrel showed back up. If our guys are going to play like they did tonight with the squirrel running around, we’ll take the squirrel."

Fans were quick to cheer their approval, while Target Field's scoreboard operator also had a little bit of fun by posting "Statcast data" on the squirrel's run.

"We should keep it at the stadium," said Nelson Cruz after his four-hit game on Tuesday. "I guess he’s part of the team now -- 26th man."

They might want to check with some of the other guys about that first.

Miguel Sanó had been in the hole when the squirrel first appeared on Monday, but he came out of the dugout and walked around the rail to avoid the critter. Mitch Garver also wanted no part of the squirrel, jumping onto the rail and straddling it to avoid contact.

"My wife's texted me about it, and she's like, 'Why are you scared of the squirrel?'" Garver said. "And I'm like, 'That squirrel is ferocious. He's trying to attack us.' He's coming at us. You should see the look in his eye. When he's sprinting at you, he's trying to get us. That's what I think. I'm paranoid about him."

It wasn't lost on Garver that the Twins lost Monday's game after the squirrel paid a visit to their dugout, while its visit to the visitors' dugout on Tuesday night spelled trouble for the White Sox in a blowout loss.

"Maybe it's not a rally squirrel," Garver said. "Maybe it's the curse of the squirrel. Whatever dugout it goes into, they lose."