That left him at 12-under 132, a stroke in front of American Jamie Lovemark (66) and Jhonattan Vegas of Venezuela (69), who both completed their rounds on Friday.
Americans Harold Varner III (67), Daniel Berger (65), Chad Collins (64), Charles Howell III (69) and Patrick Rodgers (69), and South Korean An Byeong-hun (68), were tied for fourth at eight under.
Stuard, who has twice been a runner-up on the PGA Tour but has mainly struggled on the U.S. circuit this season, was delighted to maintain his bogey-free form this week with a new putter.
"I had been having a hard time feeling like I was getting lined up right with my putts," the 33-year-old told PGA Tour radio.
"I just picked this one up last week and it felt like I could just set it down and it was right where I was looking. It's a nice feeling."
Asked about the prospect of a first PGA Tour win, Stuard replied: "Obviously it would be great, but there's a long ways to go so (I'll) just kind of focus on the next shot and see what happens."
Australian world number one Jason Day, who was also among the players who had to finish off the second round on Saturday morning after five hours of play was wiped out by thunderstorms on Thursday, carded a 68 to sit five strokes off the pace.
The cut fell at two-under 142 with England's world number 10 Justin Rose, who claimed last year's Zurich Classic title by one shot, among those who failed to advance.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Ken Ferris)
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