The fight for the job has been especially noteworthy because last year's backup, Ammon Olsen, is no longer in the picture. He transferred to Southern Utah.
Asked after the scrimmage about the situation behind Hill, Mendenhall said, "Christian Stewart is next. He's a much stronger runner than thrower, currently. However, he is a capable thrower. So, for him and for me, we're at peace and happy with where we need to be."
In 2007, Stewart threw for 3,518 yards and 44 touchdowns at nearby Timpanogos High School.
He started his college career at BYU before serving an LDS Church mission. After returning, he played at Snow College before transferring back to BYU.
In the scrimmage, Stewart completed 10 of 16 passes for 68 yards.
"The first time out running live with the No. 2 offense I felt like it went well," he said. "It was a little bit frustrating because we had a couple of good drives and then had a little mistake here, a little mistake there. [But] those things will come with time, practice and watching film."
Stewart credited Hill for helping his progress this spring.
"Taysom is a great mentor for me, even though I'm a senior," Stewart said. "He's had a year under his belt and I feel like every day we're pushing each other to get better."
Stewart's best moments came on three completions to highly-regarded junior college tight end Nick Kurtz.
"Starting camp, Nick was with the 'ones,'" Stewart said. "That's like trying to drink out of a fire hose coming into a Division I offense [with] a brand new scheme.
"So they moved him to the 'twos' and we've had a great chance to work together. … I feel like we've developed a great chemistry. I know where he's going to be and he knows I have the trust to come to him any time."
A crowd BYU officials estimated at 6,500 attended the scrimmage.
The Cougar offense scored four touchdowns and turned the ball over four times in 19 series.
Calling it "a productive day," Mendenhall singled out sophomore Michael Davis, junior Kirk Henderson and Kurtz for "… some receiver play I liked."
Davis caught a 54-yard pass from Hill. Henderson had a nine-yard touchdown catch.
Defensively, Mendenhall said sophomore safety Dallin Leavitt "continues to play well" and senior Zac Stout keeps making plays at inside linebacker, "which is fun to see."
According to Mendenhall, the job as BYU's No. 1 kicker is "about dead even" between sophomore Moose Bingham and senior Trevor Samson.
"We're charting every kick," Mendenhall said.
Tracking the Cougars
Individual statistics from BYU's scrimmage on Saturday: