Many conservatives have since concluded that if the party can get immigration off the table, Hispanics will give the GOP a new look.
Mr. Pearce agrees, but he contends that changes in policy platforms aren't enough to reverse the party's decline among voters like those in his district. Republicans must spend time in Latino neighborhoods with the respectful attentiveness of a small-town mayor.
"We have to sell ourselves," he said. It will take hard work, he added, because the majority of Hispanics are "spring-loaded" to favor the Democrats and their more expansive view of government.
Mr. Pearce's success among voters here, even those who disagree with him, underscores the hope and the difficulty of the task. Mr. Pearce said he logged more than 90,000 road miles in his district last year, a travel regimen that often separates him from his wife for weeks.
When describing his brand of constituent service in a district that is larger than the state of Florida, Mr. Pearce said, "I see myself as a big windshield wiper, just working my way back and forth, back and forth."
Bald and bespectacled, he has won the southern half of New Mexico five times since 2002. In November, he nabbed around 42% of the Hispanic vote, or nearly twice what Mitt Romney received nationally, and better than Republican Susana Martinez's share when she won the New Mexico governor's race in 2010, according to various polls.
No non-Hispanic House Republican represents a district with a higher percentage of Hispanics. Fewer than four in 10 residents of the district are Anglo, and registered Democrats outnumber Republicans by six percentage points.