When told that Rangers manager Chris Woodward was glad his team didn’t have to face him, White Sox right-hander Lucas Giolito appreciated how much everything has shifted in a year.

“It feels pretty cool,” Giolito said. “Last year, I’m sure 29 of 30 teams would’ve loved to have me for a series. It’s a little bit different now. It’s cool to see recognition from peers and other managers and things like that when it comes to the personal success I guess I’ve experienced this year. That’s a good feeling.”

With about a month left in the season, he’s in contention for even more recognition.

Giolito, who will start Tuesday against the Twins, is demonstrating that his downturn after the All-Star break was a mere lull. In his last five starts, Giolito is 3-1 with a 2.12 ERA, and he seems primed for a strong finish to his breakout season.

Who knows, it could even lead to more than a few votes for the Cy Young Award. And considering where he was a year ago, Giolito appreciates just being in the conversation.

“It’s really cool,” Giolito said. “I definitely wasn’t thinking about that going into the year. The biggest focus was making the changes I needed to make and just being a more consistent pitcher. I’ve shown I can do that and more. That’s all I concern myself with.”

Through Sunday, Giolito was in the American League top 10 in a host of categories, including ERA (sixth, 3.20), WHIP (sixth, 1.095), strikeouts per nine innings (fifth, 11.512), fielding-independent pitching (fourth, 3.20) and complete games (tied for first, three).

But burnishing his award résumé isn’t Giolito’s focus.

“I just want to take the ball every five days and win,” Giolito said. “So I know that if I’m doing that, then I’m going to put myself in a good position when it comes to numbers and everything. But, yeah, I’m not, like, actively trying to go out and win a Cy Young Award. I think that that’s something that comes as a byproduct of really good, focused work.”

At 14-6, Giolito also has a shot to win 20 games. But his view of pitcher wins is a contemporary one: They’re a team stat.

“It’s a group effort, but the way we’ve been swinging the bats, the way we’ve been playing, it’s very possible,” Giolito said. “I just need to make sure that I’m throwing shutdown innings after we score, the defense continuing to be solid behind me. We’ll see what can happen.”

Regardless of how high he finishes in the Cy Young race or if he approaches 20 wins, Giolito has shown he can shut down baseball’s top lineups. On May 23, he shut out the Astros on four hits, and he blanked the Twins on three hits last week.

That start against Minnesota, Giolito said, was the best he’d ever felt pitching. He was in control the entire game.

He challenged himself to see how much he could focus on each individual pitch and apparently measured up.

“I said, ‘If I throw 100 pitches today, I want to be 100 percent focused on 100 out of 100,’ ” Giolito said. ‘‘I ended up throwing 115, and I can say that I was focused on all of them or almost all of them. It was just one of those days where the focus was there, the execution was there. I felt really good.”