Former world number one Woods, playing his fourth US PGA Tour event since returning in January from spinal fusion surgery last April, fired a three-under par 68 for a four-under total of 138.
That was good enough for a share of the early clubhouse lead but eventually two strokes behind Canadian rookie Conners, the overnight leader who fired a 69 for 136.
Woods was almost flawless, only a bogey at his final hole, the par-four ninth at Innisbrook Resort north of Tampa, Florida, preventing him from holding the outright clubhouse lead.
"I'm up there," Woods said. "At least I'm there with a chance going into the weekend."
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That was more than some other of golf's biggest names could say, as Jordan Spieth and Rory McIlroy both missed the halfway cut.
"Today was a good day," Woods said.
"It was cold early, then the wind started blowing. It was just inconsistent, it was tough to get a bead on which way it was coming from. I did a lot of guessing -- guessed pretty good, also got fooled a few times."
In a greenside bunker at the par-five fifth, Woods blasted out to five feet and made the putt for a birdie that moved him to five-under for the tournament and put him alone atop the leaderboard.
He saved par at the seventh with an 11-foot putt, but his five-foot par attempt at his final hole lipped out.
"I didn't know I had the sole lead," Woods said. "As I was walking off nine it showed 68 and that was at four (under)."
Woods's odds to win the Masters, as adjusted by Westgate Las Vegas Superbook, dropped twice on Friday on the strength of his performance, first from 16-1 to 12-1 and then to 10-1 -- behind only world number one Dustin Johnson and reigning PGA Champion Justin Thomas, both priced at 8-1.
- 'A little bit better' -
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Woods said in January, when he returned to the PGA Tour at Torrey Pines, that the early stages of his latest comeback would all be aimed at Augusta National, where he has won four green jackets but hasn't played since 2015.
He's expected to tee it up at the Arnold Palmer Invitational next week in his final Masters tuneup.
"I keep getting a little bit better here and there, making these little subtle tweaks," said Woods, who was sidelined for all of 2016 before an abortive comeback bid last year.
"I just need to get a little bit more tournament time in. I'm starting to get a better feel for it. I'm finding the rhythm of the rounds, I'm hitting shots, I don't have a problem posting scores.