At the WGC Dell Technologies Match Play last week, he birdied on the final two holes of his first matchup with Billy Horschel to earn a draw.
"Playing with Corey, the first six holes, crazy, like every shot is right on it and then he putted good", Kim said. "That could certainly serve me well not only to the rest of this tournament, but as we look in the next few weeks".
World No 196 Conners started the day one behind Kim but he leapt into a two-shot lead after making his second birdie at the third as the South Korean made a double-bogey.
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"That was an incredible moment, I'm so lucky to be able to share this journey with her, she's there every step of the way and she's my biggest fan", Conners said.
Spieth finished second in the Texas Open in 2015.
Conners shot 66 to jump into second place on 14-under while American Charley Hoffman carded 64, the best round of the day, to vault within two shots of the lead on an afternoon that started rainy before clearing up by the back nine.
Conners, ranked 196th in the world, compiled a insane, rollercoaster six-under-par 66 that included 10 birdies in the final round at TPC San Antonio.
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Fowler's 17-foot birdie on the final hole failed to drop.
It got better on the par-four 17th, with Conners curling a 12-foot birdie putt to take him three clear heading to the 18th. He had 10 birdies and four pars that sandwiched four consecutive bogeys on the front nine.
With Sunday's final round delayed by almost two hours after heavy rain deluged the course, Conners quickly moved to the top of the leaderboard with four birdies in his opening five holes to eclipse overnight leader Kim Si-woo who struggled to get going.
The 27-year-old was able to remain at the top of the leaderboard behind 10 birdies, with four bogeys not enough to destabilise him. "Gave myself a ton of chances, hit a lot of quality iron shots set up by some good tee balls". "I'm not looking to be validated or not validated, I'm just looking for progress". "I had been hitting a lot of quality shots". "But the real reason is getting ready for next week, and I'm happy with 68". "I had a lot of confidence in my game". He was at 15 under and Connors, trying to become the first Monday qualifier to win on the PGA Tour since 2010, was a stroke back after shooting a 6-under 66. Before that, no one had done it in 24 years.
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