Clark was among a clutch of Americans high up The Open leaderboard after an opening three-under 68, two off the lead midway through the first day.
Veteran Stewart Cink, the 2009 champion, also carded a round of 68 while Jordan Spieth, who won the Claret Jug in 2017, was a shot further back with a 69 after bogeying the 18th.
Five-time major winner Brooks Koepka was also well-placed after a round of 70, along with fellow Americans Scottie Scheffler, Patrick Reed, Patrick Cantlay and Xander Schauffele.
But the resurgent Rickie Fowler, who was close to winning at Hoylake in 2014, damaged his bid with a three-over eight on the 18th to leave him with a one-over round of 72.
On a recent vacation in Italy to help recharge his batteries, Clark said that only one person had recognised him, and that was someone he had previously met.
But in calm conditions alongside the Dee Estuary, Clark made more of an impact with an impressive back nine as four birdies moved into contention to become the first person to win the U.S. and British Open back-to-back since Tiger Woods 23 years ago.
"The back nine is the tougher of the nines, I played well on the front nine but just didn't make any putts, but I had a lot of tap-in pars, which is huge," Clark told reporters.
"Then when I started making some putts, I got some good momentum."
Clark's only blot was a bogey on the 14th where his drive hit a spectator and ended in a bad position. He then found a bunker but made a superb up-and-down to only drop one shot.
"Getting a double there would have really stopped the momentum," said Clark, who added his errant ball had actually whacked the fan's iPad.
Cink may be 50 but after a blemish-free round still believes he can add to his Open victory 14 years ago.
"I have no doubts that I can win this," he said. "It's going to take a lot. It's going to take some really, really exceptional play on my behalf, but it's in there."
Spieth was temporarily knocked off track by a double-bogey six on the eighth hole and was also frustrated as a bogey at the last tarnished a superb back nine.
"I was very pleased with the round. I felt like my game has been in a really good place," Spieth said.
"I think the toughest part about the course is the crosswinds off the tee and how important it is to hit the fairways. I thought I did a really good job of that today."
Koepka birdied three of the last eight holes to make up for a slow start, including a bogey at the first.
"Just sloppy on the front. First nine holes didn't feel like I hit any good shots, but it's one of those things you've just got to find it as you go," he said.
