Reliever wanted to be in the big leagues before the birth of his first child. He achieved that goal and has been effective, with a 3.40 ERA in 29 games so far as a rookie.

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It would be hard to find anyone on the Mariners happier than Emilio Pagan.

The right-handed reliever, who was not even in big-league camp during spring training, has emerged as perhaps the team’s biggest surprise, putting up outstanding numbers.

But for Pagan, everything that has happened the past couple of months has just been a bonus.

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That’s because he had already achieved something that was much bigger to him.

“To be honest, I really only had one goal this season,” Pagan said. “My wife and I were expecting our first child, and I wanted to be up in the big leagues before she was born, and it happened. That was the only goal I had, and I reached it, so I’m happy with it.”

Pagan made his major-league debut May 3. Daughter Paxtyn, pronounced like the last name of Mariners pitcher James Paxton, was born June 14.

“When we came up with the name, I wasn’t even sure if I was ever going to be pitching for the Mariners, so I didn’t see anything wrong with it. But now guys give me a hard time about it, like, ‘Paxton’s your favorite player.’ Oh, well. It worked out.”

Just like his pitching has this season. Pagan is 1-3 with a 3.40 earned-run average in 45 innings over 29 games with the Mariners. He has an outstanding WHIP (walks and hits per inning pitched) of 0.96, the lowest for any player on the team who has pitched at least six innings.

“He really has taken advantage of his opportunity,” said Mariners manager Scott Servais. “He’s throwing the ball well. We didn’t see him a ton in spring training because he was with Puerto Rico (in the World Baseball Classic), but we saw him a little bit the spring training before. He kind of caught our eye at that point.

“He’s changed his pitching style a little bit. He’s much more aggressive, throwing a few more fastballs than he used to, and his fastball’s been really good. He’s carving a nice little niche for himself here with our team, which is great. We needed guys to step up, and he’s taken advantage of the opportunity.”

Still, the season has had its ups and downs for Pagan. Literally. He was back and forth several times between Tacoma and Seattle in May and June.

On May 23, he allowed one hit in four scoreless innings against the Washington Nationals, but he was sent back to Tacoma the next day. On June 11, he threw four no-hit innings against Toronto, then was sent back to Tacoma the next day.

On each occasion, the pitching-strapped Mariners could not afford to keep a reliever who would be unavailable to pitch for a few days.

“We explained to everybody in (Class) AAA that we were in kind of a revolving-door situation,” Servais said. “They haven’t necessarily liked it, but they’ve understood it. I think they appreciated our honesty with them.”

Pagan didn’t let the demotions get him down.

“You want to see the good work you do get noticed, and it was,” Pagan said. “They told me I did great and they appreciated all the work I was putting in. But you also know the situation. We had a ton of injuries. So it’s nothing to be bitter about, upset about. I was fortunate and blessed to have an opportunity, and I’m very thankful.”

Pagan, 26, was a 10th-round draft pick in 2013 out of Belmont Abbey, an NCAA Division II school in North Carolina. Drafted No. 297 overall, he is the highest draft pick in the history of the school, just ahead of former big-league outfielder Alex Castellanos.

Pagan was also a third baseman at Belmont Abbey. He often played at third base for eight innings, then moved to pitcher in the eighth or ninth to close out games. As a junior, Pagan had 13 saves and did not allow a run in 21 innings.

Pagan had another good season pitching and hitting as a senior. He hit nearly .300 with four homers in each of his two seasons at Belmont Abbey, but he knew his future was pitching.

“Of course, I miss hitting,” he said. “Everybody loves hitting, but hitting’s really hard, and I couldn’t hit anything off-speed. I could hit the fastball, but that’s it, so I knew if I was going to have a chance to play professionally it would be as a pitcher. The pitchers up here are really good, so I don’t know if I’m missing too much.”

Pagan, who was born in Simpsonville, S.C., made steady progress through the Mariners’ organization. But with Pagan on Puerto Rico’s team for the World Baseball Classic, he was a bit of an afterthought at Mariners spring training.

With the team’s rash of injuries, though, Pagan is one of many pitchers who got unexpected opportunities this season. Perhaps no one has seized that opportunity more than he has.

“The one that really rose up was Emilio Pagan,” said Mariners general manager Jerry Dipoto. “He was a guy through the years in the organization that always kind of endeared himself to everybody. The fact that … he was added to a WBC roster, we didn’t even bring him to big-league camp, believing he was going to miss most of it with the WBC. Lo and behold, he wound up pitching a critical role for us for much of the season.”