Four strokes back from the leader at the start of play, Spieth went six-over 42 on the front nine, which included two double-bogeys and two bogeys without a birdie.
Graeme McDowell, who arrived in Texas on the back of a first win since 2015 in the Corales Puntacana Resort and Club Championship, lies three shots off the lead after an opening 69 matched by Ireland's Seamus Power.
Kim Si-woo on Thursday reeled off five birdies in a six-hole stretch en route to the first-round lead at the Valero Texas Open in San Antonio.
Hoffman, the 2016 Texas Open victor, shot 67 for 18-under on the weekend.
Spieth, who is without a top-10 finish since last year's Open Championship, was part of a large group a shot further back after five birdies and a solitary bogey in his 68.
American Brian Stuard carded a 66 to share fourth place with Kim on 15 under, one ahead of compatriot Kevin Streelman, who briefly held a share of the lead on 15 under when he made a fourth successive birdie at the 17th before he bogeyed the last to sign for a 64.
After a drive to the left side of the fairway on the 481-yard par-4 fourth hole, Conners had a 157 yard approach shot, setting himself up for the birdie.
He had a one-stroke lead when he rammed home a 34-foot birdie putt at No. 12, and stayed a stroke up with a tap-in birdie at the 14th.
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At the 183-yard par-3 16th, Conners hit a tee shot 175 yards at the green, setting himself up for the 5-foot putt for birdie. He birdied four of the first five holes at TPC San Antonio. This time, the ball rolled past the hole to inside four feet for what looked like an easy birdie.
He remained on track with a birdie at the 14th after getting up and down from a greenside bunker and, after holing a clutch 12-footer for par at the 15th, he notched another birdie at the par-three 16th where he almost took out the flag with his tee shot. Later in the day, Kim was grabbing at his right upper body with an apparent injury.
Kim surged to a two-shot lead when he chipped to a foot away on the par-5 14th hole and tapped in for birdie.
The other bogey, on a par-3 at No. 7, he hit his tee shot into a bunker, then blasted over the green.
"It's always nice to get out and make some birdies early, shake off some rust", Fowler said.
Jordan Spieth has not recorded a top ten on the PGA Tour since the Open Championship past year but this week, back on a golf course on which he finished runner-up the last time he played the event, he is within striking distance heading into the weekend.
Both must win the event to earn an invitation to the first major of the year and join countrymen Jason Day, Marc Leishman, Cameron Smith and Adam Scott at Augusta.