WASHINGTON — Gov. Rick Perry of Texas on Sunday sent some of his clearest signals of interest in a 2016 presidential bid, joking wryly about his “botched” run in 2012 but then adding that “I think America is a place that believes in second chances.”

Appearing on the NBC program “Meet the Press,” Mr. Perry defended his state’s use of the death penalty as “appropriate and humane,” even after the bungled execution in Oklahoma last week.

He called for a greater national focus on job creation and the underemployed, particularly women. And he criticized President Obama for a “one size fits all” approach to solving problems that are perhaps best left to the states.

But as Mr. Perry begins traveling across the country and abroad to extol the vigorous Texas economy and raise his national profile, he chose not to deflect discussions about his presidential aspirations.