MOBILE, Ala.—Jim Zeigler didn’t have much time to celebrate.

The morning after his Republican primary victory on Tuesday and a gathering of supporters at Wintzell’s Oyster House, the 78-year-old hit the road early to drive the nearly four hours to his hometown of Sylacauga, southwest of Birmingham, to attend a funeral.

But even this early, Zeigler already had his mind on November.

“It’s going to be a tough, tough race,” Zeigler said.

He may be right.