Tiger poised to strike at US Open leader Thompson
"I'm just going to go out and do my best," the 27-year-old Thompson said of his second-round strategy.
"It's a US Open, I could get bad bounces. That kind of stuff is out of my control. But I can build on the confidence that I gained (Thursday) and go out and have fun."
With the help of seven birdies, Thompson posted a four-under par 66 on Thursday, giving himself a three-shot lead over Woods and four others.
The chasing group at The Olympic Club included 2010 US Open champion Graeme McDowell, England's Justin Rose, 2001 USPGA Champion David Toms and Nick Watney -- who propelled himself into the title hunt with the third albatross in US Open history.
Meanwhile a host of expected contenders found themselves in a battle just to make the halfway cut.
Phil Mickelson, with five runner-up finishes in the national championship he would dearly love to win, opened with a six-over 76, while Masters champion Bubba Watson posted an eight-over 78.
Northern Ireland's Rory McIlroy, who missed just 10 greens all week en route to a record-setting US Open triumph at rain-softened Congressional last year, missed 12 in a seven-over 77, while world number one Luke Donald's bid for a first major title evaporated with a nine-over 79 on Olympic's treacherous hillside Lake Course.
McIlroy and Donald teed off at 7:44 am along with world number three Lee Westwood, just trying to salvage something in the second major of the year.
"I've got to hit it in the fairway for a start," McIlroy said of the task ahead. "I've got to hit it onto the green.
"You just keep it simple and try to hit fairways and try to hit greens and not be too greedy."
Woods followed that recipe to perfection on Thursday.
"He struck it really well," said Mickelson, who was paired with Woods and Watson in what was billed as a super-group for the first two rounds.
"Had really solid control of his flight, trajectory. The way it occurred, it was impressive."
Woods has endured an erratic season -- winning two USPGA titles to end a tour-level victory drought that stretched back to 2009 but also pulling out of one tournament with an injury and finishing 40th in the Masters.
He was efficient on Thursday, pulling out his driver just three times and often relying on irons off the tee.
He hit 10 of 14 fairways and 11 greens in regulation, and three of his misses were tee shots that rolled off the slanted fairways into the first cut of rough.
The task facing him as he teed off with Mickelson and Watson at 1:18pm (2018 GMT) was just to keep it up.
"I was able to put it together in a major championship," said Woods, whose pursuit of Jack Nicklaus's record of 18 major titles has been stalled at 14 since he won the 2008 US Open at Torrey Pines.
"I'm going to need it the next three days."