The 32-year-old from Connecticut, in only his fifth start on tour, was among 14 players unable to finish before darkness.
Stegmaeir, on 11-under, heads fellow Americans Chad Campbell (67), Morgan Hoffman (66) and Tyler Aldridge (68) by one stroke.
Aldridge, a journeyman who has played most of his competitive golf on satellite tours in the United States and Canada, nearly holed out with his second shot at the par-four 18th, tapping in from six inches for his third birdie in the final four holes.
He was delighted to be in contention for a maiden PGA Tour title after playing just one full season on the U.S. circuit, back in 2009.
"I'm very pleased," the 31-year-old told Golf Channel.
"I am hitting the ball well, putting well and things are going my way. It feels great (to be back on the PGA Tour).
"It's been a while and I've put a lot of hard work in. I'm just looking forward to what's to come."
American world number five Rickie Fowler, the highest-ranked player in the field, opened with a one-over 72 on Thursday, but bounced back with a 65 that featured an 80-foot putt for eagle at the 13th to move within six shots of the lead.
"It's nice to have a tee time tomorrow," Fowler said after ensuring he would make the cut.
"I didn’t hit the centre of the (club) face a whole lot yesterday. (Today I) made a couple of good swings, got things going, made a few putts."
Big-hitting compatriot Finau had the shot of the day when he holed out with his second shot from 235 yards at the par-five 13th.
Finau was not far away from a second albatross two holes later when he drove the green at the par-four 15th where his ball landed a few feet left of the pin.
He birdied the hole, but double-bogeyed the next and, with one hole left, will miss the cut unless he birdies the 18th hole on Saturday morning.
(Reporting by Mark Lamport-Stokes in Los Angeles; and Andrew Both in Cary, North Carolina; Editing by Ian Ransom)
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