TAOS, N.M. — Voters may wonder whether the fire in Gary Johnson’s belly burns hottest for recreation. If he doesn’t win the presidency, the Libertarian Party nominee plans to ski 120 days next year.

His support in the polls is a little shy of 10 percent, suggesting he’ll get that chance. Yet Mr. Johnson, the former Republican governor of New Mexico, is hoping to have an impact, if only by hastening the fall of his old party and the rise of his new one as a national political force.

“That’s going to be the consequence of what we do, at a minimum,” Mr. Johnson said in a breakfast interview near his mountain home here.

He wears bicycle gear after riding 14 miles uphill to reach the restaurant. For Mr. Johnson, that’s a respite from the cross-country media blitz he hopes will lift him to the 15 percent threshold required to qualify for presidential debates with the Democratic and Republican nominees, Hillary Clinton and Donald J. Trump, beginning next month.