Swafford recorded three birdies on each nine to post an 11-under total of 133, finishing a stroke in front of compatriot Cameron Tringale (65) and Zimbabwean Brendon de Jonge (70).
American Boo Weekley, co-leader overnight with de Jonge after opening with a 64, was at 11 under with three holes remaining when play was suspended for the day due to the threat of approaching thunderstorms.
The horn sounded at 4.30 p.m. local (2130 GMT) to halt the action and the entire afternoon wave of players will be back in position on Saturday morning to complete the second round
Swafford, who has recorded just one top-10 in 16 starts on the 2014-15 PGA Tour, was delighted to maintain his improved form after struggling earlier this year.
"Nothing was really clicking for me so I kind of changed the takeaway with my golf swing, and actually my stroke with the putter," Swafford, who missed six cuts in the first three months this year before tweaking his putting stroke, told Golf Channel.
"I've been trying to bring it a little more inside and down the line. Ever since then, Bay Hill, it has kind of kicked in and I've really seen some good golf."
Swafford revised his putting stroke in late March before the Arnold Palmer Invitational at Bay Hill where he tied for 11th place.
Burly Zimbabwean de Jonge, seeking his first PGA victory this week on his 225th career start, carded three birdies and a lone bogey in the second round despite the receptive conditions.
"I've just got to stay aggressive," said the 34-year-old.
"Obviously there's going to be a bunch of birdies this weekend and the golf course isn't going to be able to firm up.
"I am excited about it. I feel like my game is in good shape so hopefully keep it going."
The cut was projected to fall at four-under 140 with U.S. Ryder Cup players Rickie Fowler and Steve Stricker in danger of missing out.
(Reporting by Mark Lamport-Stokes in Los Angeles; Editing by Frank Pingue/Steve Keating)
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