Masters champion Scott, winner here in 2011, fired a sparkling four-under-par 66 on another warm, blustery day at Firestone Country Club to finish at three-under 207.
Though that left the Australian a distant 12 strokes behind the pacesetting Woods, he was a more respectable five adrift of Swede Henrik Stenson in the battle for runner-up honours in the elite World Golf Championships event.
"I'm going to have to be really hot tomorrow, but it was a good round out there today," world number five Scott told reporters after mixing five birdies with a lone bogey at the challenging par-four last.
"Anything in the sixties is a good score around here no matter what the conditions are. There's always plenty of trouble. My game is sharpening up to where I want it to be, so I'm happy about that so far."
After seven-times Firestone champion Woods had spread-eagled the field with a scorching 61 in Friday's second round, Scott instantly revised his goal for the week.
"I'd like to have a very hot weekend and maybe finish second," Scott, 33, said with a broad grin.
So far, so good for the nine-times PGA Tour winner, who is also building momentum for next week's PGA Championship, the year's final major.
"It'll be nice to play another solid round tomorrow," Scott said. "I was nearly bogey-free today, just dropped that one on the last, and that's good golf around here.
"It's also good to get those rounds in before a major next week where dropped shots are so costly."
Asked to assess how well his preparations had gone for the PGA Championship at Oak Hill, Scott replied: "I think I've done what I needed to do the last couple days.
"I'm going to go and hit some more balls tonight and keep the good feelings. I've been driving it a little better since Thursday, and I feel it's getting to where I would like it to be. I think by next Thursday I can be really sharp."
(Reporting by Mark Lamport-Stokes; Editing by Gene Cherry)
