Mallon left an 18-foot, downhill birdie attempt about 3 1/2 feet short on the last hole, squandering a chance to match the LPGA record of 59 recorded by Annika Sorenstam in Phoenix two years ago.
'It's funny saying you're disappointed to shoot 60, but when you get that close, it's pretty exciting,' Mallon said.
Mallon started on the back nine of the cozy, 6,176-yard Dell Urich municipal course and birdied Nos. 10, 12-14 and 16 for a 30. On her back nine, Mallon made birdies at Nos. 3, 4, 5, 6, and 8.
On her 16th hole, the seventh, Mallon's 6-foot birdie putt curled away. She reached 10 under for the round and 16 under for the tournament on the next hole with a 12-foot birdie putt that barely reached the cup.
Mallon opened a three-shot lead over first-round leader Lorie Kane early in the second round.
Mallon, who has won three majors and 11 overall titles, played with Sorenstam when she shot a 59 at Moon Valley Country Club in 2001.
'I thought about her a lot,' the 39-year-old Mallon said. 'That day, the ball was rolling in from everywhere. Of course, she birdied the first eight holes, so it was a little different. And you have to give her credit - that was a par-72. She was 13-under.'
Vicki Fergon (1984), Sorenstam (1999) and Karrie Webb (2000) all had 11-under rounds that are ahead of Mallon's score in relation to par.
Sorenstam, Webb and Se Ri Pak (1998) were the only women to shoot 61s in LPGA competition until this tournament. Kane had a 61 on Thursday, and Norwegian rookie Suzann Pettersen followed her opening 72 with a 61 Friday.
The Urich course is being used as the Tucson LPGA venue for the first time. It replaced the par-72 Randolph Park North course, where the course record was a 62 by Kristi Albers in 1998.