CHICAGO — When the calendar flipped to July, Gerrit Cole received a fresh slate to get back on track following a rocky June.

The Yankees veteran right-hander is now about to enter August on a heater.

Cole delivered the goods Tuesday to finish off a strong July in style, tossing six shutout innings to lift the Yankees to a 3-2 win over the White Sox at Rate Field.

Aside from a fourth inning in which he briefly lost his command, Cole was sharp in his 12th start back from Tommy John surgery, scattering four hits and two walks while striking out seven and even making a circus catch on a pop-up to help himself out.

“My instincts are sharper, my reads are better, I’m in the flow of things, I’m getting pitch ideas or pitch locations that are proving to be the right ones,” Cole said after pitching to a 2.97 ERA in five July starts, on the heels of a 6.12 ERA in five June starts. “That comes with sharpening up all facets of the game. Bringing the stuff in tighter, bringing the location in tighter. And to be honest, our catchers have been great. They deserve a lot of credit for the run prevention.”

After Angel Chivilli and Brent Headrick kept the shutout alive in relief of Cole, David Bednar — pitching for the fourth time in five days — made things interesting in the ninth when he gave up a two-out, two-run homer to Chase Meidroth.

The runs were the first he allowed since May 18, snapping a 21-game scoreless streak, but Bednar held on to finish off the White Sox (55-51).

It might not feel like it given their injuries and a mostly struggling lineup, but the Yankees (61-46) have won 11 of their past 15 games.

With the trade deadline looming, they now sit just 1 ¹/₂ games behind the Rays for first place in the AL East.

“We’re trying to play as clean and as crisp baseball as we can in all facets of the game,” Cole said. “Keep trying to improve, keep trying to push ourselves. Just keep pushing, pushing, pushing to get a little bit better. We know we have some guys that are injured, but at the same time, that shouldn’t affect the way we go about our business and the way we try to play our brand of baseball. I think we’ve been committed to that and it’s paid off for us.”

A night after an offensive outburst, the Yankees scratched across three runs in the second inning — on run-scoring doubles from Anthony Volpe and Max Schuemann — which were just enough in the end.

Cole’s six strong innings were sorely needed after the last two Yankees starters — Max Fried and Will Warren — had lasted a combined six innings, putting an extra toll on the bullpen. His 99th and final pitch of the night was a 97 mph fastball — a pitch that has continued to get better with each passing start — that got Braden Montgomery whiffing for Cole’s seventh strikeout to finish the sixth.

“I just feel like he’s getting in more and more of a rhythm of getting his competitive sea legs again and getting into the routine every fifth and sixth day,” manager Aaron Boone said.

Cole added some comedic relief to his outing in the bottom of the second inning, right after being handed the 3-0 lead, when Colson Montgomery skied a pop-up in front of the mound.

Catcher Austin Wells initially made an attempt on it and even stepped on Cole’s foot while backing up before the pitcher called him off and took it himself. The ball popped out of his glove, but Cole recovered and snagged it with his bare right hand as he backpedaled onto the mound.

Once the ball was secure, Cole held out his arms and called Wells for pass interference on the play.

“I started to work back into the ball after I called it and Austin stepped on my foot,” Cole said. “I thought I was cooked, thought I was going down, but luckily I made some juggling act and saved it.”

The Yankees also got some terrific defense (much more traditionally than Cole’s) from Jazz Chisholm Jr. at second base while Chivilli, on his 24th birthday, continued to look like a reliever Boone can trust heading down the stretch — the latest recipe for finding a way to win.

“I think it’s becoming part of our identity and we’re taking pride in it,” Volpe said. “That’s all that matters. As long as we get the job done, we’re just going to keep working on that.”