When Cory Gardner was persuaded by national Republican leaders to run for the Senate in Colorado against incumbent Democrat Mark Udall, he was a latecomer to the race. Mr. Gardner was a one-term House member and the 2014 midterm election was eight months away. And it was soon discovered from a U.S. Chamber of Commerce poll that Mr. Gardner had a problem: Colorado’s population is 22% Hispanic and the poll found that Mr. Gardner was supported by only 11% of Hispanic voters, a dismal showing.

Mr. Gardner says he was “unknown...