LIVONIA, Mich. — Four days before the South Carolina primary, Gov. John Kasich stepped off his campaign bus into a crowded Republican Party office near Detroit on Tuesday morning and marveled at how many people had shown up to see him.

“You think I got free stuff in that bus?” he joked. But he turned serious before leaving. “This state,” he told the crowd, “is going to be very important to me.”

After focusing on New Hampshire and emerging with the success story he was hoping for, a second-place finish, Mr. Kasich is now embarking upon a new and similarly critical quest: to sell himself to the people of Michigan, which neighbors Ohio, the state he governs.

After holding four town-hall-style meetings in Michigan this week, Mr. Kasich returned to South Carolina on Tuesday afternoon, and he is not setting that state aside. But he sees his bigger test as Michigan, which votes on March 8, a week after Super Tuesday, which includes many primaries in the South.