On May 13, the Rays were mired in mediocrity: a .500 team at 19-19, sitting three games back of the Red Sox in the American League East, on the cusp of being swept at home by the Yankees.

Rich Hill was on the mound that day for a win that started one of the best stretches in team history. It’s only fitting, then, that Hill was on the mound in New York on Monday as the Rays earned a 3-1 win over the Yankees to stay hot and improve to 35-20. Tampa Bay’s 22-6 record in May is the best in the Majors this year, and it sets a franchise record for wins in a calendar month.

“Everything came together,” said Rays manager Kevin Cash, echoing a familiar theme amid his team’s 16-1 run. “We played a really complete game.”

The backbone of that game was Hill, who faced the Yankees for the third time this season. New York hit him hard in the early stages, knocking seven balls in play with exit velocities exceeding 100 mph. But Hill managed to scatter three hits and a pair of walks over five scoreless frames on an economical 56 pitches.

Hill’s biggest pitch might’ve come in the fifth inning with one on, one out and leadoff man DJ LeMahieu at the plate. Hill sunk an 0-2 curveball low in the zone, and LeMahieu hit it to a diving Yandy Díaz at third base. Díaz started an inning-ending double play, halting one of the Yankees’ few offensive threats on the day.

“[Hill] is not the biggest ground-ball guy, but he gets the ball on the ground [and] Yandy makes a tremendous play,” Cash said. “To me, that was the play of the game.”

Offensively, the Rays put their leadoff man aboard in every inning from the third through the sixth, and three of them came around to score. That included Austin Meadows, whose 334-foot fly ball crept over the short porch in right field for a solo shot.

Meadows is now 20-for-59 (.339) in his past 15 games, with 13 extra-base hits and 20 RBIs. Surely, he’s not going to complain about the generous dimensions of Yankee Stadium, the only park in the Majors that wouldn’t have contained Meadows’ drive, according to Statcast.

“We benefited, probably, from the elements of the stadium a little bit with Meadows,” Cash said. “But like [Meadows] said, ‘I’ll take it, whatever.’ He got a cutter and knocked it out of the ballpark.”

That homer in the fourth inning provided the eventual winning run. Once Hill exited after the fifth, a quartet of relievers held the Yankees to two hits, a walk and one run the rest of the way. So, again, the Rays won due to a strong starter, a dash of power from the offense, a key defensive play and strong bullpen work. A complete game.

Hill was at the center of it, though, which is fitting given that he was at his best all month long, just like the Rays. After posting a 7.25 ERA in five April starts, the 41-year-old lefty turned his season around with a 0.78 ERA in six May outings spanning 34 2/3 innings.

“Another day, another dollar, another great start from Rich Hill,” center fielder Kevin Kiermaier said. “Just incredible what he can do. ... I don’t know how he does it, but I do know he works his butt off.”

With Hill having kept his pitch count low through five strong innings, there was reason to wonder why he was given the hook so early. Cash said that Hill was not dealing with any injury or illness, and that the Rays simply wanted to give the Yankees “a different look” by that point in the game.

Hill understands that, and he’s bought into the Rays’ occasionally unconventional roadmap to 27 outs. It’s that sort of team-first mindset that has fueled their current ascent, dating back to May 13.

“Everybody’s pulling for each other as much as they’d be pulling for themselves in any situation,” Hill said. “As long as we continue to keep doing that, that’s something that has proved to be a winning formula.”