NEW YORK — After opting out of his minor-league deal with the Red Sox last month, right-hander Adam Ottavino was biding his time, waiting for the right opportunity to present itself.

He was confident that he is still a big-league pitcher, not wanting to sign with a club and pitch in Triple-A, serving as a someone’s backup.

So, when the Yankees came calling, with closer Devin Williams going on the paternity list, it was a no-brainer for the reliever to put the pinstripes back on.

He signed a big-league deal with the team and got to the ballpark on Tuesday afternoon, his first time back with the Yankees since the two-year stint he had with the club from 2019 to 2020.

“We tried with the Yankees for a while, even this offseason,” Ottavino said. “I felt like I was kind of in a little bit of a no man’s land for the last week or so. But it all came together pretty quickly. Obviously, I’m grateful for the opportunity. I’m very lucky for it.”

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Ottavino, 39, admitted that his stuff was down and not impressive enough to pitch in the big leagues early in spring training with Boston. He’s entering his 15th MLB season and it’s tough to build up quickly, especially when he signed with the Red Sox midway through February. But Ottavino threw the ball well right before he exercised his opt-out and the Yankees noticed that uptick in his effectiveness.

“He’s been on our board a little bit, I know the front office has been talking about him the last couple weeks,” Yankees manager Aaron Boone said. “Felt like what he was doing at the back end of spring training was in line with who Otto is. With Devin, felt like an opportunity to get him in here and excited to have him back.”

To Boone, Ottavino has “a lot left in there.” He referenced the right-hander’s track record and the fact that he’s been pitching in big markets for much of this past decade.

“He’s still been very effective throughout his entire career,” Boone said. “When we first got him, we brought him in to be kind of that righty killer, high-leverage set-up, and he delivered on that. Obviously, very familiar with with what it takes to play here, and he’s played here and the Mets and Boston. This is what he knows. So there’s a chance to get a quality pitcher in here and hopefully he can help us. Where it goes, we’ll see.”

Ottavino is well aware that this could be a short reunion. Williams won’t miss too much time and as of now, all the veteran knows is that the Yankees wanted to sign him for Tuesday’s game. The only other realistic option would be for the Yankees to option Brent Headrick down to Triple-A, but even if they do that, relievers on the injured list like Ian Hamilton aren’t far away from rejoining the active roster.

He’s not looking beyond this first game back in a Yankees uniform regardless. Ottavino has grown to understand that in this game, tomorrow is never guaranteed.

“Regardless of however it plays out, it’s thrilling and really exciting,” said Ottavino, a native of New York City. “It’s a team that I obviously love and a lot of love for everybody in here. Just excited and feeling lucky, just trying to enjoy it.”

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Max Goodman may be reached at [email protected].