The American world number one carded a second successive one-under 70 in strengthening winds at Royal Troon to finish at two-under 140 - five strokes behind leader Shane Lowry.
Although his caddie Ted Scott was struggling with an upset stomach, Scheffler made several key putts to save par on the back nine and finish in good position to challenge for his third major title.
"I definitely made a few more putts than yesterday," Scheffler, who has won six times on the PGA Tour this year in 15 starts, told reporters. "The golf course was more challenging today than yesterday.
"I holed a good amount of putts. Being able to hole a few, especially some important ones on the back nine for par, those always feel good, keeping the momentum."
Scheffler offset a birdie at the fourth with a bogey at the fifth before picking up two more shots at 14 and 16 and recording another bogey at the last.
He is now looking to emulate fellow American Arnold Palmer, who won the Masters and five other tournaments on the PGA Tour in 1962 before going on to win that year's Open at Royal Troon.
"I've played two solid rounds and it put me five shots back, and I'll continue to try to execute and just continue to try to hit good shots and hit good putts out there," Scheffler.
"So far it's been two really successful days, keeping my head in it nice and taking the good breaks and the bad breaks in stride and just continuing to grind. That's what's most important to me."
Scheffler paid tribute to his caddie Scott.
"He had a bit of food poisoning last night ... or some sort of stomach bug," Scheffler said.
"But he seemed to feel better as the day went on. You saw him laying down probably quite a bit out there, which was pretty funny. He's a fighter. He did a good job today and battled through it."