Both Mr. Paul’s words on Tuesday and the scene that played out on stage before he spoke showed how he is working to get black voters to give him a closer look. He pledged to repeal “any law that disproportionately incarcerates people of color,” and he invoked the Rev. Dr. Martin Luther King Jr.’s description of “two Americas,” one rich with opportunity, the other marked by “daily ugliness.” A black pastor from Louisville preceded him to the stage, as did J. C. Watts, who at one time was the only black Republican in Congress. The audience, however, was overwhelmingly white.

For some time, Mr. Paul has been trying to establish himself in places where Republicans do not often find receptive audiences: at historically black campuses like Howard University, in liberal college towns like Berkeley, Calif., and Austin, Tex., and in the moneyed corridors of Silicon Valley.

At the same time — and providing his critics with fodder to label him a political contortionist — Mr. Paul has sought to remain in the good graces of the social conservatives, Tea Party fiscal hawks and libertarians who helped propel him to an improbable Senate victory in 2010, when he was little-known and had never run for office.

Image A Rand Paul supporter at his announcement on Tuesday. Credit... William DeShazer for The New York Times

He offered his conservative fans reassurance on Tuesday that he is no liberal, saying: “If we nominate a candidate who is simply ‘Democrat Lite,’ what’s the point? Why bother?”

Mr. Paul did not mention that he opposes abortion rights and same-sex marriage.

He shares those views with his father, Ron Paul, the former Texas congressman who attained folk-hero status among libertarians and some conservatives through his campaigns for the nomination in 2008 and 2012.

But Mr. Paul has never seemed entirely comfortable being known in politics as his father’s son. He has not fully embraced the higher-proof libertarianism that endeared Ron Paul to so many of his die-hard followers. And in turn, they have not fully embraced him.