Even if he defeats Hillary Clinton in both Iowa and New Hampshire next month, Senator Bernie Sanders faces a daunting problem when the presidential race moves on to bigger, more diverse states: winning over black voters.

Starting this weekend in South Carolina, he is trying to solve it.

With a blitz of appearances, ads on black-oriented radio stations, a tour of historically black colleges and the help of well-known and not-so-well-known African-Americans, Mr. Sanders is racing to get the word out: He is a lifelong civil rights advocate who marched with the Rev. Dr. Martin Luther King Jr.

He talks substantively about issues vital to many blacks, like “radically rethinking police procedures,” even in front of all-white crowds. And the economic policies that he relentlessly argues would combat income inequality and injustice across the board, he says, apply all the more to persistent racial disparities in American life.

“We have a real tragedy with youth unemployment,” Mr. Sanders thundered last weekend in Des Moines, where the only black faces in the crowd of 1,500 appeared to be those of journalists following him. “Kids who graduated high school who are white: 33 percent unemployed or underemployed. Latino: 36 percent. African-American: 51 percent.”