Spieth shot a 66 to post a seven-under-par 133 halfway total and is tied for third at seven-under with Patrick Reed, who played just eight holes on Friday.
Half the field did not complete the second round due to a five-hour morning thunderstorm delay, with some managing only six holes.
Spieth, who started at the 10th, picked up four shots in five holes after the turn, including birdies at the difficult fourth and fifth.
"That was a big back nine to climb back into contention. I was very pleased with that run there,” Spieth told Golf Channel.
The world number two added that he was shaping 90 percent of his shots with a draw, moving the ball from right to left, to increase his “comfort level”.
“That’s important for me right now, to know where the ball is going to start and if it’s going to move one direction.”
In his third tournament since he frittered away the Masters with a quadruple bogey at the 12th hole during the final round, Spieth sounded upbeat with the U.S. Open three weeks away.
“I was set up by fairways hit,” he said. “The greens were two feet faster than yesterday and it took me a little while to adjust but once I made that adjustment (the birdies followed).”
He was particularly satisfied to play the difficult three-hole stretch from No. 3 dubbed the “Horrible Horseshoe” in two under par.
The second round will resume on Saturday at 7:30 AM local time (1330 BST), with the third scheduled to start early afternoon.
Molder looked set to take a handy lead to bed after five front-nine birdies, but a double-bogey at the par-four 12th brought him back to the field.
Clubhouse leader Simpson has had to adjust to a regular-length putter since ditching his long stick last year ahead of the Jan. 1 “anchoring” ban and expressed satisfaction with his work on the greens.
“I’ve switched putters and been putting great the past couple of days,” said the 2012 U.S. Open champion.
“The good thing about me and the bad thing about me is once I get confidence, I seem to be really confident, and once I get a little down, I seem to be get really down. But through two days it’s been solid, so I’m excited.”
(Reporting by Andrew Both in Cary, North Carolina; Editing by Peter Rutherford)
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