The 46-year-old, who began his second round on the 10th tee, carded five birdies across his opening nine holes before adding another on the front nine. He completed 36 holes without dropping a shot, sealing his round by sinking a 12-foot par putt at the last to reach 14 under par.
Glover leads fellow American Lee Hodges, who signed for a 66 to sit two shots back on 12 under, while Zac Blair is a further stroke adrift after closing with a 68.
Glover's PGA Tour exemption earned by his back-to-back victories in 2023 is due to expire later this year, making a strong finish this week particularly valuable. However, the veteran insisted his focus remains simply on playing good golf.
"I'm too old to chase that stuff, to be fair," Glover said. "I just want to play. I just wanted to play well again, and I've struggled. Having two days like this just gives me a little belief that it's in there, you know? Been working hard at it and hadn't seen many results and have for two days. Doesn't mean anything tomorrow. Doesn't mean anything Sunday.
"But for these first two days my work has been paying off," he said. "I got a couple more days to keep fighting and keep doing it."
Another veteran, 50-year-old Zach Johnson, also remained in contention after posting a second-round 70. The Iowa native, who opted to play his hometown event instead of the U.S. Senior Open, was tied for 11th at eight under, six shots off the lead heading into the weekend.
The cut fell at three-under 141, with Jordan Spieth surviving on the number despite carding three bogeys over a five-hole stretch on the back nine in a round of 69.