Mr. Paul and those close to him are confident that his die-hard libertarian-leaning supporters will not desert him, and that gives him freedom to build bridges beyond that base.

“He’s becoming a translator between the grass-roots conservatives and the establishment,” said Trygve Olson, a consultant who bridges the two wings. He then added an implicit dig at other Republicans: “He’s actually demonstrating leadership.”

Transcending the two worlds can be tricky: While Mr. Paul voted with Mr. Cruz on the effort to defund the health care law — pre-empting future primary attacks from the right — he also said publicly over the summer that he thought shutting the government down was “a dumb idea.” Privately, he complained during the shutdown that the effort was futile and was damaging the party.

Still, he is clearly the beneficiary of the comparison with Mr. Cruz: Establishment Republicans are lining up to heap praise on Mr. Paul, using words like “grown” and “matured” to describe him and the role he played during the shutdown.

“This ordeal showed a side of Rand that I though was politically very smart in terms of his tone and trying to distance himself from a strategy that clearly didn’t play well for us,” Senator Lindsey Graham, Republican of South Carolina, said in an interview.

The standoff over health care and the shutdown also highlighted the personal differences between the two men and how they are viewed within the Senate: Mr. Paul is more easygoing and speaks casually as he makes his points with fellow senators. Mr. Cruz, his colleagues complain, often seems like he is lecturing them — or, as one put it, “still on Hannity’s show.” While Mr. Paul was overheard on a hot mike plotting strategy with Mr. McConnell, Mr. Cruz was receiving tongue-lashings from his Republican colleagues at private senators-only luncheons. Mr. Paul mixes with a range of senators at the weekly Republican luncheons; Mr. Cruz tends to stick close to his fellow hard-liner Senator Mike Lee of Utah.

Nowhere is the competition between the men more obvious than in the crucial state of Iowa, where Mr. Paul was the most sought-after speaker in the state earlier this year in the aftermath of his filibuster over the use of drone strikes, but where Mr. Cruz is now surging after his starring role in the shutdown battle. He was the headline speaker at the Iowa Republican Party dinner late last month.