Beau Hossler is once again knocking on the door.
The day began with Irishman Dunne taking the first round lead as Thursday's weather-hit first round was completed.
But Hossler put himself back in the fold to break his duck in Houston, despite bogeying his opening hole.
Spieth lost in a playoff in Houston in 2015 and went on to win the first of his three major titles the following week at Augusta National.
He turned pro after his junior year, but Hossler was sidelined immediately because of surgery due to a torn labrum. "I just got really dizzy". It's the second time in as many months Hossler holds at least a share of the lead after 36 holes, the last time coming at the AT&T Pebble Beach Pro-Am. He said patience will be key as he seeks his first PGA Tour victory. It would be his first appearance there or in any major outside his three U.S. Open showings.
Paul Dunne of Ireland, who finished his first-round 64 on Friday morning, was 9 under after a 71.
Dunne had been awake at 4.45am local time in preparation for a 7.20am restart to his round but then had just 30-minutes before heading into his second round.
Tour rookie Sam Ryder was 8-under through 15 holes and was on top of the leaderboard. But he rebounded with a 4-under 68 to move to a tie for second at 10 under.
After a promising start to his 2017-'18 campaign, Rickie Fowler has gone quiet for a while.
Fowler blistered the outward half with four birdies and added another at 13 to briefly join the lead before a poor drive at the last found water and led to a bogey.
After turning in 33, he missed four long-range birdie putts at the start of his homeward nine before finally seeing a 25-footer disappear for his fourth birdie of the day at the 454-yard fifth.
Madison resident Steve Stricker needed to win the tournament to earn a berth in next week's Masters, but missed the cut - the line was 3-under - by a stroke.
Fowler, who last won in capturing last December's Hero World Challenge in the Bahamas, then pared his next four holes only to drop a shot at the last after sending his tee shot into water down the entire left side of the final hole for his score of a four-under 68. Follow all the action on Golfweek.com and our Facebook and Twitter feeds.