DETROIT -- The Tigers introduced Eduardo Rodriguez on Monday in a mid-afternoon press conference at Comerica Park. Yet manager A.J. Hinch still had his mind on breakfast with Carlos Correa.

“I had a ham and cheese omelet,” Hinch said of last week’s surreptitiously photographed meeting with the free-agent shortstop in Houston. “I didn’t order lunch, so it was only breakfast. And I didn’t realize it was as big a deal as it was. He’s having a baby, [so I was] delivering a baby gift.”

Give Hinch credit for leaning into the storyline that seemed to hover over the event. While the Tigers were saying hello to their newest pitcher, at least as many questions at Monday’s press conference revolved around the Tigers’ pursuit of a shortstop.

As aggressively as the Tigers have attacked their offseason to-do list, adding a workhorse starter and trading for a catcher, they still have their biggest void at shortstop. Whether they can land Correa, or one of the other highly valued shortstops on the market, remains a mystery, one that seems at least partly by design.

“We’re actively looking for a shortstop,” Avila said. “I don’t know where it’s going to take us, but we’re going to try.”

The uncertainty is not for lack of trying, and not for throwing all their efforts at Correa. Avila has reached out to agents for all of the major shortstops -- not just Correa, but Corey Seager, Marcus Semien, Trevor Story, Javier Báez and Dodgers superutility great and part-time shortstop Chris Taylor. Much of that took place at MLB’s General Managers Meetings earlier this month, which Avila and Hinch attended together.

Beyond that, Avila and Hinch have been traversing the country on recruiting trips, trying to sell free agents of various positions on the idea that Detroit is an up-and-coming team ready to take the next step with a revived fan base.

“There’s a ton of interest in coming to Detroit by a variety of players,” Hinch said, “and that’s because I think people are aware that our trajectory is going in the right direction. So [we’ve had] great conversations, not just the one [with Correa] that was photographed, some really good meals Al and I have been sharing.”

Tigers owner Christopher Ilitch was asked three times at Monday’s press conference about the possibility of signing a top free-agent shortstop. Each time, he stated his support for the long-term plan Avila, Hinch and the rest of Tigers management have mapped out.

“We’ll have all the resources we need as a ballclub to ensure we continue to improve,” Ilitch said, “and ultimately to be playoff contenders and World Series champions. That’s our goal. Al has had a plan from Day 1, and he’s been executing his plan very, very well.”

Ilitch was asked essentially the same question in a slightly different way: Can the Tigers still be aggressive, however the offseason unfolds?

“Al’s going to have the resources that he needs to go out and accomplish what he thinks we need to accomplish in the offseason,” Ilitch said. “He and I talk about contracts and different contract scenarios quite often. We’ve got an open mind. Anything is really on the table as it goes regarding that.

“Having said that, we also want to make sure that what we do going forward allows the organization sustainable success over the long haul, and so we need to keep that in mind.”

Avila did not want to enter a discussion of what kind of deal fits sustainability, saying each club has to make its own decisions on payroll allocation. But he frequently reiterated points he made before the offseason began.

“In our situation, obviously shortstop is a need,” Avila said. “I’ve mentioned where one big splash does not win you the offseason. It’s making some smart moves along the way and putting together a good, sustainable roster.”

Thus, while Avila talked about the obvious need at shortstop, he also referenced alternative plans to signing a top free agent, from possibly trading for a shortstop to adding an offensive boost in the outfield.

“There’s different ways of putting together a good 26-man roster,” Avila said. “If it doesn’t work out at the shortstop position, then you have to fortify your team in other ways.”

Even in that scenario, however, the Tigers would still likely need a shortstop addition on some level, whether a secondary free agent or a trade. The latter becomes more viable if another team signs a top free-agent shortstop and suddenly has a former shortstop to deal.

“It’s possible,” Avila said of a trade. “It’s the hardest way of doing it, obviously, but it’s possible.”

Perhaps by coincidence, Avila noted that shortstop prospect Ryan Kreidler, the Tigers’ No. 10 prospect according to MLB Pipeline, has been mentioned by other clubs in trade discussions.

“I’ve mentioned that I’ll be very hesitant to trade any players that we feel are going to have an impact very soon or immediate future,” Avila said.

The Tigers know trading for a shortstop is hard; they explored it when the Rays made Willy Adames available last spring, only to be outflanked by the Brewers. Realistically, all of the Tigers’ choices for upgrading at shortstop are a challenge, just in different ways.

The Tigers will have to find a way to do so, whether it’s in the next couple weeks or later in the offseason.

“We’re engaged in different conversations with agents and other clubs and possible trades, and we’ll continue to work,” Avila said.