"A.J. gets on the phone and calls the traveling secretary for the Padres and says, 'Get me a plane ticket out of here tonight,'" Cora said.
"You're leaving already?" Cora asked.
"Yeah, this kid won't last seven picks," Hinch said.
The Astros selected Correa with the first pick, taken by a brand new general manager, Jeff Luhnow, who had just begun a massive reconstruction of the organization.
When Correa made his Major League debut three years later at 20, his arrival signaled that the reconstruction was almost complete. The Astros have reached the posteason in two of the last three years and will play Game 1 of the American League Championship Series presented by Camping World on Friday.
As for Hinch and Cora, they exchanged telephone numbers during that trip and agreed to stay in touch. Cora turned down some feelers from the Astros in both 2013 and '15. Finally, last season, he was ready and accepted an offer to become Hinch's bench coach.
"It had not felt right until then," Cora said. "I had an offer from ESPN, which was very convenient for me. Instead of being away from the family the whole season, I was going to be home half the season and doing baseball. I think ESPN helped me get better at what I'm doing now."
Cora has gotten rave reviews from Hinch and Luhnow, with both saying he's ready to be a Major League manager at a time when there are openings with the Red Sox, Tigers, Mets and Phillies.
"He's driven to do well, has very high expectations of himself, which in turn gives high expectations to the people around him," Hinch said. "He has a little bit of an 'it' factor when it comes to having convictions in his decisions. I'm proud of him because he has been able to influence players very, very quickly."
Astros third baseman Alex Bregman mentioned a conversation with Cora moments before hitting the game-tying home run in Monday's decisive Game 4, a 5-4 victory over the Red Sox at Fenway Park in the AL Division Series presented by Doosan.
"He came up to me and said, 'Hey, one at-bat, that's all that matters,'" Bregman said. "I was 0-for-10 going into that at-bat. And he said, 'One at-bat, that's it. Have some fun, play the game how you did the beginning of the series.'"
Cora makes it clear that he'd like to manage. He also makes it clear that he has another priority with the Astros right now.
"I'm very honored that people are mentioning my name," Cora said. "Wow, I never thought it would be like this. But right now, THIS is exciting. Playing Game 4 at Fenway was exciting. I can't wait for the next opportunity this team has. Not because of me, but because of those guys.
"You see Alex Bregman going crazy. Seeing Carlos Beltran hitting the ball off the wall [in ALDS Game 4]. That was so cool. Carlos and I played against each other in Connie Mack [Stadium]. We played together with the Mets. Now I'm coaching him. Stuff like that gets me excited right now just talking about it. I'm having a blast, and we'll see what happens after that."
That Cora would eventually return to baseball feels like a foregone conclusion. He grew up in and around the game in Puerto Rico. His late father, Jose, founded the Little League chapter in Caguas, and his older brother, Joey, played 11 seasons in the Majors.
"Breakfast, lunch, dinner, it was baseball in my family," Cora said. "Obviously, I was blessed with a brother that went through everything. Everything that Joey did was what I wanted to do. My hero was sleeping next to me and was eating rice and beans just like me. I didn't have to look for somebody else.
"You talk to my mom about baseball, she's passionate about it. My 14-year-old daughter, she knows what's going on with the Astros. She was at the World Baseball Classic. She laughed and cried and suffered with them. That's the cool thing about it."
Cora played for six organizations over 14 seasons. He served as both GM and manager of the Puerto Rican national team, and at one point fired his manager -- himself.
"I tried to do both jobs," Cora said. "That was a little much."
During his playing career, Cora played for Terry Francona, Davey Johnson, Jim Tracy and a long list of others. He was part of two teams that went to the World Series, then did four years of analysis work for ESPN.
Cora sees this season as Hinch's bench coach as part of his baseball education.
"I'd had three managerial interviews -- Rangers, Padres, D-backs -- and everybody mentioned my lack of experience," Cora said. "I was like, 'If this is one of my goals in life and if lack of experience is what's holding me back, I can take care of that.'
"It was a perfect fit. Young team. Young manager that can relate. Everybody knows how the Houston Astros do business. This was a perfect opportunity and a chance to win. You look at the talent they have, and you think, 'Man, this is a good gig.'
"I played  years in the big leagues. I wasn't the most talented. I had to pay attention to be successful. I had to think things through. It's something that intrigues me."
His job with the Astros was part of a season that has seen Cora experience the highest of highs and the lowest of lows. First, there was Puerto Rico advancing to the Classic final. Then came Hurricane Maria sweeping through Puerto Rico and causing massive damage.
"The first thought of the day is our island and the last one, too," Cora said, referring to Beltran, Correa and other Puerto Rican-born players and staff with the Astros. "Our guys are playing for more than the team they're representing. They're playing with a heavy heart, but they're playing with a lot of passion for their country."
Long Day. Tough to be away from my love ones and countrymen. Que el Se��or nos colme de calma y paciencia. Se que nos recuperaremos. 💯%🇵
Thanks to Astros owner Jim Crane offering a plane, Cora was able to get his mom, Iris, and several family members off the island. But whenever this season ends, he'll return to Puerto Rico to assist in cleanup efforts.
"Sixteen percent of the island had electricity yesterday," Cora said, "and today it's 11 percent. It's like one step forward, two steps back. It's becoming tougher and tougher. Joey was there last week. When he got back to Pittsburgh, he called me and said, 'Be ready.' This is when we're at our best -- everyone helping each other. I see it in Instagram.
"Ricky Martin went down there. Marc Anthony. Jennifer Lopez. Lin-Manuel Miranda. They live in the States, but they're going back there and rolling up their sleeves and helping. They're not just helping in San Juan. They're helping in the mountains where there's no communication, no water, no nothing. That's what I want to do."
That'll happen when the Astros are done, and Cora's hoping that won't be for a few more weeks.
"I've loved it here," Cora said. "This is fun. Not only because we're a winning team and have a winning organization, but because the coaching staff, the advance scouts, everybody works together. Every day I come here, there's something there to learn. That's what I appreciate."
Richard Justice has been a reporter for MLB.com since 2011. Read his columns and follow him on Twitter at @RichardJustice. This story was not subject to the approval of Major League Baseball or its clubs.