Playing in the morning wave on the hilly, tree-lined course, Kim birdied two of the first three holes, then made a big move on the back nine with five birdies in a seven-hole stretch.
"I just want to keep moving forward the next two days," the 33-year-old South Korean player said. "I try to move forward whatever my position."
Australia's Hannah Green, the 2023 and 2024 winner at Wilshire Country Club, was 8 under after a 69 and six shots off the pace.
Green has three worldwide wins this year — the HSBC Women's World Championship on the LPGA Tour along with the Women's Australian Open and Australian WPGA.
First-round leader Chizzy Iwai of Japan was tied for second with Ina Yoon of South Korea.
Iwai followed her opening 63 with a 68, making three birdies in a four-hole stretch on the back nine late in the afternoon.
She's also looking to attack the course.
"I like aggressive golf," Iwai said. "I think tomorrow I need to be aggressive."
Yoon had a 64. She played her opening nine holes — the back nine at El Caballero — in 7-under 29 and added a birdie on the par-5 seventh.
The Chevron Championship is next week in Houston, with the Dallas area-based Kim excited about her strong play leading into the major.
"I'm very much looking to forward next week," Kim said. "But I have two more days, so I just focus on this weekend."
Kim won the BMW Ladies Championship last year in South Korea for the last of her 13 LPGA Tour titles. She won the 2020 KPMG Women's PGA Championship for her lone major title.
Jin Hee Im (68), Melanie Green (68) and Jessica Porvasnik (69) were five strokes back at 9 under.
The tournament is being played at El Caballero for the second straight year because of course renovations at Wilshire.
Defending champion Ingrid Lindblad was 3 under after a 72.
Meanwhile, Australia's World No. 6, Minjee Lee, withdrew from the LA Championship prior to the start of the second round. No reason was provided by the LPGA Tour.
Lee, 29, was tied for 40th after she posted a three-under-par 69 in Thursday's first round.
