The young Australian is four shots adrift of leader Jeeno Thitikul (66), at 12 under for the tournament, and three behind New Zealander Lydia Ko.
Kyriacou's five-under round comprised an eagle on the eighth, five birdies and two bogeys.
The 23-year-old Sydneysider had 24 putts on Friday and hit 11 out of 18 fairways.
She is in pursuit of Thitikul, who zoomed right past the field with three birdies over her last four holes for another six-under 66.
Ko played bogey-free with an eagle on the par-5 eighth hole for a 66. Thitikul played in the afternoon, made birdie on all the par 5s at the TPC River's Bend and finished strong.
Thitikul hit pitching wedge over the water and avoided going long, setting up one final birdie that put her at 12-under 132.
Haeran Ryu, who won the FM Championship three weeks ago for her first LPGA title, matched the low round of the day with a 65 and was two shots behind.
World No.1 Nelly Korda (70) is tied 15th and five shots behind at seven under.
Thitikul is coming up on the two-year anniversary of her last individual victory, after teaming up with Ruoning Yin to win the Dow Championship this year.
"Just looking forward for good golf and enjoyable golf out there," Thitikul said. "I think a lot of good vibes coming from me, starting from like Olympics. ... I just want to play my game. Just want to control on my side, not others."
Ko has good vibes from a summer she won't forget. She won the gold medal at the Olympics, a victory that got her into the LPGA Hall of Fame. And then she won the Women's British Open for the second time, this one at St Andrews.
Key to Ko's round was her eagle on No.8, and Ko appeared to be the least likely to do that. Rose Zhang hit her second shot into short range. Charley Hull had 15 feet for eagle. Ko was off the green hopeful of getting up-and-down for birdie. Instead, she holed it.
"I kind of felt left out because Charley hit it to like 15 feet for eagle; Rose hit it to like two feet; and I had missed the green," Ko said. "I hit it exactly the way that I was envisioning. As soon as it came off the club face and landed, Charley said, 'Good shot.' And five seconds later it went in the hole."
Australians Robyn Choi (69) is at five under with Grace Kim (70) a shot behind. Defending champion Minjee Lee (71) and Gabriela Ruffels (69) are at two under.
Australian Hira Naveed (70) missed the one-under cut by one after back-to-back bogeys on 13 and 14 cost her dearly.
Notables to miss the cut include Rose Zhang (even par) and Danielle Kang (one over).
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