The Cruz playbook calls for drawing support in three of the four voter blocs within the GOP. Writing off the party’s establishment wing, he is playing to win the GOP’s tea-party activists, who are his natural constituency; libertarians, a smaller group that he has been trying to peel off from rival Sen. Rand Paul of Kentucky; and evangelicals.

His biggest challenge, supporters say, is introducing himself to evangelical voters who know little about him. The campaign and one of the super PACs supporting him have been trying to spotlight his work defending religious liberty before he was elected to the Senate. A flier distributed at the Bob Jones University rally detailed court cases he argued before the Supreme Court, including a case involving the display of the Ten Commandments at the Texas capital. Some of the super PAC ads took the same tack.

His campaign is counting on the idea that the more evangelicals know about him—and about Mr. Carson—the more likely they are to back Mr. Cruz. People at Bob Jones University, where the two candidates appeared a day apart, were able to compare their night-and-day styles, with Mr. Cruz offering as fiery a presentation as Mr. Carson’s was low key.

The comparison worked to Mr. Cruz’s advantage in the case of Dee Dee Groves, who attended both events undecided between the two and came out a Cruz supporter. “I really appreciate his strong Christian stand,” said Ms. Groves, who is home schooling her children in Greenville. “I still like Carson, (but) I feel like he’s not dynamic enough.’’