Mr. Cotton, 39, is also trying to seduce early-state Republicans while playing nice with his colleagues.
He was the keynote speaker in March at the Silver Elephant Dinner for rising stars among South Carolina Republicans, and he won a speaking role at the Iowa Economic Forum lunch with convention delegates here.
Mr. Cotton was one of five freshman senators who traveled to Arizona in support of Senator John McCain, who is facing a tough re-election fight there. He attended a Republican dinner in Missouri for Senator Roy Blunt, who is also in a tough race.
In August, Mr. Cotton will travel the country, including a stop in Nevada to campaign for Representative Joe Heck, a Republican, who is running for the seat being vacated by Senator Harry Reid, the minority leader.
Like Mr. Cruz, Mr. Cotton is attempting to establish himself on the right flank of the potential primary field over one of the most galvanizing issues among conservative activists. “I think the fact of the matter is that right now, we have too much immigration,” Mr. Cotton said.