Chris Murphy broke through the gridlock on guns. He’s not done.

Clad in running shoes and a UConn baseball cap, Sen. Chris Murphy walks along the shoulder of a two-lane road, a red umbrella bobbing above his head. Occasionally, he’s splashed by a passing car. 

The junior senator is on his sixth “Walk Across Connecticut,” in which he treks from one end of his state to the other, talking with voters and taking selfies along the way. It’s the kind of event that politicians often invite the press to cover. But Mr. Murphy, his team warns, likes to walk alone.

For...