Champ already won the Sanderson Farms Championship two weeks ago, held opposite a World Golf Championships event. He was leading by one shot when his tee shot sailed right and landed in bushes near rocks.
What put Kuchar in good shape were the five straight birdies on the front nine, and then he took the lead for the first time with a birdie on No. 11 before his wild adventure on the 12th hole.
Jordan Spieth was likely done for the year.
Kuchar was at 14-under 128 as he tries to end more than four years without a PGA Tour victory.
Champ missed only two greens for the second straight round at El Camaleon Golf Club, including his final one that led to a lone bogey.
"If you're not driving it well, this course is extremely hard", said the seven-times PGA Tour victor.
"I knew Sybi was in the crowd, I knew she had caddied for me before and was friendly with Zach Johnson, and figured she would be good to handle it, and Zach would go easy on her. Just to be able to place my ball right in the fairways and on the greens and make the putts when I needed is great".
Hickock, a buddy of three-time major victor Jordan Spieth, is playing in just his eighth PGA Tour event and his fourth this season after spending previous year on the developmental Web.com tour, where he won the DAP Championship and had five other top-10 finishes. He had a 69 and was outside the cut line with half of the field still on the course in good scoring conditions.
2 share lead at Tour Champions season finale: PHOENIX - Paul Goydos and Tim Petrovic shot 8-under 63 to share the lead in the Charles Schwab Cup Championship, leaving points leader Bernhard Langer seven strokes back in the PGA Tour Champions' season finale. Next up is his wedding over Thanksgiving weekend.
Bozzelli lost ground with a four-under par 67 that left him in a group of seven players sharing third place on 131, a group that also included New Zealand's Danny Lee and India's Anirban Lahiri. Eighteen players were within five shots of the lead.