The 26-year-old, chasing her first LPGA title, rebounded with a birdie at the 11th before posting further gains on the 16th and 17th to finish eight-under heading into the weekend.
Olson has built a reputation for thriving under the pressure of majors, having recorded a tied-ninth finish at ANA Inspiration earlier this year.
"I love majors. I love it when par is a great score on some holes, and if you can sneak a birdie or two out, that's great," Olson said.
"But I just love the atmosphere. I love that it's difficult and it really rewards good and penalizes bad shots.
"Fortunately I've gotten some good bounces, been stroking it pretty well, and rolling ball well."
Overnight Puerto Rican leader Torres recovered from consecutive bogies in wet conditions to sign for 69, while both South Korea's Lee and American Martin carded rounds of 66.
Carlota Ciganda, who shared the opening round lead with Torres, struggled with her putter on the front nine but a 70 meant the Spaniard stays one shot behind the leaders.
Last month's Women's British Open champion Georgia Hall finished with a second straight 68 to stay in a seven-strong group at six-under overall.
(Reporting by Hardik Vyas in Bengaluru; Editing by Christian Radnedge)
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