Conners carded a 3-under-par 68 on Saturday to claim a one stroke lead at 9-under 204 after the third round at Innisbrook Resort's Copperhead Course in Palm Harbor, Fla.
"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 past year.
The 42-year-old, making just his fourth PGA Tour start since back surgery, bounced back from a blemish at the 13th to birdie the next and safely negotiated the Snake Pit to remain well-placed to challenge for a first worldwide victory since 2013.
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Woods has showed some signs of better performance lately, finishing 12th in his most recent start, at the Honda Classic last month.
PGA Tour rookie Conners only got into the event as an alternate but leads the way after carding a 69 for a six-under 136 total.
Woods will play in the penultimate group on Sunday alongside Snedeker who also carded a 67 while Rose fired a five-under 66 - highlighted by an eagle at the 11th - to reach eight-under.
"I'm starting to see him get into a pretty good groove out there and hit the shots he's trying to hit", said Joe LaCava, Woods' caddie.
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Chelsea are in danger of going four games without victory having won just four of their last 15 fixtures in all competitions. Where and how to watch the live TV coverage of Premier League game? "It's a pity that at the end we conceded the goal ".
Corey Conners has no business being here, but he hardly cares.
"I've played with Tiger many times". About the time he was dropping a shot, Conners was across the lake at No. 12 and rolled in a 20-foot birdie putt to stretch his lead to two shots over Rose, three shots over Woods. This will be the fifth time Snedeker has played with Woods this year. The crowd roared their approval and Tiger gave them a fist pump and a tip of the cap. "I don't know what my future holds for me". He didn't fare much better on Friday with two birdies, two bogeys, a 71 and a phone call to get back to Dallas sooner than he wanted. "Every putt I had I thought it was going to go in".
"Could I have envisioned myself being here?"
Conners still settled himself with a wedge he almost holed on No. 9 for a birdie of his own. "Just kind of vintage Tiger". He hit 14 greens Saturday and got up and down to save par (or better) three of four times. "Yeah, first couple days there's sketchy stuff tee to green, like my 1st Round".
Live updates from Tiger Woods' Round 2 at the Valspar Championship
Larrazabal was tied for second place with Paul Peterson and Matteo Manassero, two shots behind leader Emiliano Grillo. And that challenge was real, as playing partners Henrik Stenson and Jordan Spieth would attest. "I'm revved up.
Graeme McDowell made the cut on the number after a one-over 72 left him on three over.