The 15-time major champion fired a roller-coaster two-over par 74 to share 19th place on one-over 145 after 36 holes at Augusta National, nine adrift of leader Scottie Scheffler but only four back of second place.
“I’ve got a chance going into the weekend,” Woods said. “Hopefully I’ll have one of those lightbulb moments, turn it on and get it done.
“On the back nine on Sunday anything can happen. I’ve just got to get myself there.”
World number one Scheffler, seeking his first major title, matched the low round of the day of 67 to stand on 136 with a five-stroke lead, equalling the largest 36-hole edge in Masters history.
“I wouldn’t say much changes,” said Scheffler. “I played solid golf. I put myself in position. I have to just keep doing what I’m doing.”
A second-place pack included South African Charl Schwartzel, the 2011 Masters champion, 18-hole leader Im Sung-jae of South Korea, Ireland’s Shane Lowry and defending champion Hideki Matsuyama.
“It was one of my better rounds of my career,” said 2019 British Open champion Lowry. “It was quite tricky out there, quite difficult.”
Woods, back in action 14 months after a horrific car crash, teed off only three strokes behind the leaders but made four bogeys on the first five holes.
“It was blustery. It was windy, swirling all over the place,” Woods said. “I got a couple bad gusts and I made a couple bad swings.
“Then on four, I ended up in a divot. It was almost, ‘What else could go wrong?’”
He was at risk of missing the cut despite a solid opening 71, but rallied to carry his comeback charge into the weekend as he chased a record-tying sixth Masters crown.
“I got back in the ball game,” Woods said. “Tomorrow is going to be tough.
“It’s going to be quicker, drier, faster, and it will be a great test.”
Woods will rely upon therapists and ice baths to control leg swelling and get himself ready to play again.
“My team has done a hell of a job getting the body ready,” Woods said. “After I go ahead and break it out there, they go ahead and repair it at night.”
After the woeful start, Woods chipped to two feet and tapped in to birdie the par-5 eighth, saved par from six feet at nine and tapped in from three feet to birdie the 10th.
Woods missed an eight-foot par putt at the 11th then found a bunker at the par-3 12th and missed a 12-foot par putt.
He responded by making a three-foot birdie putt at the par-5 13th, curling in a nine-foot birdie at 14 then closing with four pars to make a 22nd consecutive Masters cut.
Even as Woods struggled, the 46-year-old legend drew throngs of fans.
After being hospitalized for weeks and unable to walk for months, rehabilitation work has allowed the medical marvel to make an epic comeback in an iconic setting, although his gait is affected by a right ankle no longer working well.
“It’s never going to move like it used to,” he said.
McIlroy 10 back
Woods, ranked 973rd, teed off in the third-to-last group alongside only Chile’s Joaquin Niemann after 2010 British Open champion Louis Oosthuizen of South Africa withdrew due to injury.
He missed seven-foot par putts at the first and third holes, missed the green at the par-3 fourth on the way to bogey and found trees off the tee to bogey the fifth.
Four-time major winner Rory McIlroy, seeking a green jacket to complete a career Grand Slam, fired a second straight 73 to stand on 146. McIlroy hasn’t won a major title since 2014.
Those missing the cut on 148 included four-time major winner Brooks Koepka and three-time major winner Jordan Spieth on 150, Tokyo Olympic champion Xander Schauffele on 151 and 2020 US Open winner Bryson DeChambeau on 156.