The American shot 24 places up the ranks after a staggering round saw him end at 12-under on the day, giving him a real shot at his first PGA Tour win in 120 starts.
Cole, though, has Ryan Gerard hot on his heels after a 68 left him one shot off the lead, while England's Jordan Smith - who led after two rounds - dropped five strokes off the top after a pedestrian 74 featuring five bogeys.
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"It was a great day," Cole said after his round. "Everything was kind of working well - I got off to a really good start.
"I saw, I kind of knew going into it that it was going to be a harder course, firmer conditions. Getting off to that start and realising how much harder the course was today was great.
"Then, yeah, just kept trying to hit good shots one after the next and make birdies when you can.
"It's not going to be an easy day tomorrow, I know that from my experience.
"I know that it's going to be difficult, but that's why I practice really hard and that's why I try and do everything the way I do so that I could be as prepared for whatever tomorrow brings."
JJ Spaun and Mac Meisner are still in contention in the final round at 10-under, even with the former making three bogeys in his third-round 68.
Alex Smalley, Hideki Matsuyama and Russell Henley are all three strokes off the lead and tied in fifth, with Matsuyama - the only non-American in the top five - making one birdie and one bogey in his 300th tour start.
European Ryder Cup star Ludvig Aberg highlights the rest of the field, sitting five strokes off the lead after making four birdies and three bogeys in a 69.
Watch the final round of the Charles Schwab Challenge on Sky Sports. Get Sky Sports or stream with no contract.
