The 19-year-old Spieth, playing on a sponsor's exemption this week, fired a five-under-par 66 in relatively calm conditions on a challenging layout to finish at seven-under 135, alongside overnight leader Roberto Castro, who carded a 69.
South Korea's Lee Dong-hwan was at five under after shooting a seven-birdie 66, level with Argentina's Andres Romero, who still had five holes to complete.
Play was suspended at 2:44 p.m. local time due to the threat of lightning and an attempt late in the day for the action to resume had to be aborted as storm clouds once again swept across the area.
The second round will continue on Saturday morning before the cut is made and the players then begin the third round in threesomes with a two-tee start, officials said.
Spieth, competing on a limited US PGA Tour schedule this year as a special temporary member, was delighted with his bogey-free display, highlighted by five birdies in his first nine holes.
"I hit 18 greens ... I was very solid from tee to green," the former University of Texas standout told Golf Channel. "I put myself in great positions off the tee and was able to hit my mid-irons a little better than yesterday.
"Every opportunity I get (being) in contention, I learn a little bit each time," said Spieth, who has posted four top-10s in 13 starts on the 2013 US Tour, his best finish a tie for second at the Puerto Rico Open.
"I felt the nerves on the back nine but was able to still strike the ball well."
Spieth, who turned professional in December, has yet to win his first US Tour title but hoped to draw on memories of his college success while bidding for victory at Congressional over the weekend.
"I just need to approach it like the last tournament that I won, and I am thinking back to college and just the way that I was able to win out there," the 19-year-old recalled.
"Obviously I don't know what it's like to win on the tour yet, but I just can't really think about that. I am only halfway through the tournament."
Castro, like Spieth seeking a maiden victory on the US Tour, was happy to remain in contention after following his opening 66 with a 69.
"Just got to keep going, and I was able to do that today," the flame-haired 28-year-old said. "It feels good to play two rounds under par. It's a very hard golf course."
Americans Cameron Tringale (67) and James Driscoll (69) were at four under, a stroke better than Swede David Lingmerth, who was among a group of four players after firing a best-of-the-day 65.
Australian world number four and Masters champion Adam Scott, the highest-ranked player in the field, was a distant nine strokes off the pace after returning a 71.
The cut was projected to fall at two-over 144 with former US Tour winners Charley Hoffman, Aaron Baddeley, K.J. Choi and Hunter Mahan among those in danger of missing out.
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