The 31-year-old from Michigan ended his round in champion style, hitting a superb hybrid three-iron from 214 yards to just two feet for a tap-in putt as he posted a 10-under total of 130.
That left Stuard one stroke in front of Australian Marc Leishman (64) and Japan's Hideto Tanihara (65), with American Harris English (66) a further shot back at eight under.
"I feel comfortable on the greens," Stuard, who has twice finished second on the PGA Tour in pursuit of a maiden victory, told reporters. "I feel like I read them pretty well and I'm able to make putts."
Australian world number two Scott, the highest-ranked player in a surprisingly strong field for the PGA Tour's first full-field event of the year, lurks just three strokes off the pace after carding a five-birdie 66.
Scott, whose regular caddie Steve Williams is back in his native New Zealand this week for an auto race competition, almost holed out for eagle at the last, his ball hitting the flagstick before bouncing down just six inches from the cup.
That shot was one of many that stunned Scott's substitute caddie, surfer Benji Weatherley, who had never previously carried a bag in a professional tournament until this week.
"It's the most fun you could ever have," Weatherley told Golf Channel about working for Scott at Waialae this week. "I've no nerves because, for one, he's so good it's embarrassing.
"Every single shot is what you see (as a highlight) on Sports Center, especially that last one. It was unbelievable."
SURFING INTRODUCTION
Scott, who was initially introduced to Weatherley by multiple ASP World Tour Champion surfer Kelly Slater four years ago in San Diego, has been equally impressed this week by his stand-in caddie.
"Benji has been unbelievable," said Scott, who ended the round in a six-way tie for fifth, level with first-round leader Bae Sang-moon of South Korea (70). "He's done so good and he's really getting a knack for it.
"He talked me down on (hole) 18 today. He's like, 'It's not driver, it's three-wood.' So he put me in the fairway there. He's really getting the hang of it.
"We're having a lot of fun but he's taking it serious and he knows it's serious to me. It's a fun opportunity, I think, for both of us."
World number seven Zach Johnson, who clinched his 11th PGA Tour title at the Hyundai Tournament of Champions in Kapalua on Monday, carded a 67 to end the second round five strokes off the pace.
Also at five under was fellow American James Hahn, who produced the shot of the day by holing out from 193 yards in the right rough for a rare albatross (three-under par) at the par-five ninth.
It was the first albatross of the PGA Tour's 2013-14 season and helped Hahn to a 68.
The cut fell at one-under 139 with former world number one Vijay Singh, 2003 Masters champion Mike Weir and Japan's Ryo Ishikawa among those missing out.
Jordan Spieth, the 2013 PGA Tour's Rookie of the Year and a runner-up last week at the Hyundai Tournament of Champions, also failed to advance after carding a 71.
Due to a forecast for poor weather on Saturday, tee times have been brought forward for the third round.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Peter Rutherford)
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