The Yankees acquired lefthander Jaime Garcia and cash from the Twins for two prospects on Sunday morning — and they might not be done trying to improve their rotation ahead of Monday’s 4 p.m. non-waiver trade deadline.

Even after picking up Garcia, the Yankees continued their efforts to land righthander Sonny Gray from the A’s. Gray is the top available starter on the trade market and the Yankees are considered the favorites to grab him in a deal for prospects.

In Garcia, the Yankees picked up a veteran who already had been traded once this season, from the Braves to the Twins last Monday. Garcia, 31, is a combined 5-7 with a 4.29 ERA. He will be a free agent after the season.

“It’s a good pickup,” third baseman Todd Frazier said. “A real good pickup.”

The Yankees sent pitching prospects Zack Littell and Dietrich Enns to the Twins. Littell, 21, was 14-1 with a 1.87 ERA in Class A and Double-A ball. Enns, 26, was 1-1, 2.19 for Triple-A Scranton/Wilkes-Barre.

The small-market Twins sent an undisclosed amount of cash to the Yankees to help cover the approximately $4.5 million left on Garcia’s contract.

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Manager Joe Girardi said Garcia likely will make his first Yankees start on Thursday in Cleveland.

“He’s been talked about for a while,” Girardi said. “He’s an experienced starter. He’s a guy who’s pitched in some high-leverage situations. He’s pitched in St. Louis, where there’s been big expectations. We’re excited that we got him.”

Garcia pitched in parts of eight seasons with the Cardinals, winning 13 games in both 2010 and 2011. He suffered through some injury-plagued seasons before returning to win 10 games in both 2015 and 2016. Garcia, who is 67-52 with a 3.65 ERA in nine seasons, signed with the Braves in the offseason.

“Basically a three-pitch guy,” pitching coach Larry Rothschild said. “Fastball, changeup, slider. Pretty good movement on his fastball. He’s started a lot of games and been through pennant races and he knows how to pitch. He’ll help us.”

The Yankees had been using lefthander Caleb Smith as their No. 5 starter, but he failed to get out of the fourth inning in both of his starts and was sent down to Scranton on Sunday.

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Gray, 27, who is 6-5 with a 3.43 ERA, was scratched from his scheduled start Sunday in anticipation of a trade somewhere. He is signed through 2019, which means the A’s can try to get a bevy of prospects from interested suitors. They also are believed to be shopping Gray to the Dodgers, Astros and Cubs, among others. Another available top starter is Yu Darvish of the Rangers, but he will be a free agent after the season.

If the Yankees add another starter, they will have six. Girardi said he doubts he would go to a six-man rotation.

The Yankees are monitoring the workloads of young starters Jordan Montgomery and Luis Severino, so they could choose to hold one or both of them back from a start or two to shave innings before the end of the season. Girardi and Rothschild, however, said there are no firm innings limits for the young duo.

The Yankees’ clubhouse was buoyed by the news of the Garcia trade, which was completed at about 9 a.m. The players know there may be more to come.

“We have until [Monday], too,” said Frazier, a recent acquisition himself. “We’ll see who else we’re going to get.”