The 26-year-old needed to birdie his last three holes to become the sixth player to dip under 60 on the US Tour but he lost momentum as he finished par-birdie-bogey on another day of ideal scoring conditions in the California desert.
Chappell came agonisingly close to chipping in for birdie at the par-four seventh, his 16th hole of the day, and very nearly chipped in for eagle from greenside rough at the driveable par-four eighth.
Needing to hole out from the fairway with a nine-iron from 164 yards at the ninth to card a 59, he pushed his approach 40 feet from the pin and three-putted for his only bogey of the day.
"I hit a good putt," Chappell told reporters about his long-range birdie attempt on the ninth green. "Unfortunately I missed the come-back putt but, all in all, it was a good day.
"I got off to a great start. I chipped in on the first hole, birdied the next two and that just kind of got me going. And two par-fives on that backside (back nine) helps.
"I made some beautiful putts and really wedged it well today. I hit my numbers and was able to get up and down," added the American whose best US Tour finish was a runner-up spot at the 2011 Texas Open.
Chappell, who ended up in a tie for eighth on Sunday at 22-under 266, raced to the turn in a blistering six-under 30 and further birdies at the first, second, fourth and sixth got him thinking about a possible 59.
"I hit it on David Duval's plaque on 18," he said of his drive at the par-five 18th, referring to the 59 shot by fellow American Duval in the final round of the 1999 Bob Hope Classic.
"And I said, 'Well, David Duval made it from here.' I probably hit it to right where he hit that putt from. The pin's a little bit further left, and then I hit a mediocre eagle putt, but I tapped it in for birdie.
"I just wanted to keep hitting good shots and stick to my process and see what it added up to."
After failing to birdie his penultimate hole, Chappell knew his chance of firing a rare 59 had realistically gone.
"It wasn't a very good number for me," he said of his approach into the ninth green. "Even with adrenaline going, I had 164 yards to the hole and that's a huge nine-iron.
"I have to turn it right-to-left to get it back there, and last thing I want to do is overturn it. But I hit it where I'm supposed to hit it, I guess, and then just I hit a bad second putt. What can I do?"
The five players who have shot 59s on the PGA Tour are Americans Al Geiberger (1977 Memphis Classic), Chip Beck (1991 Las Vegas Invitational), Duval, Paul Goydos (2010 John Deere Classic) and Australian Stuart Appleby (2010 Greenbrier Classic).