SAN JUAN, P.R. — José Andrés was walking along a dark street in a stained T-shirt and a ball cap, trying to decompress after another day of feeding an island that has been largely without electricity since Hurricane Maria hit a month ago.

He’d gone barely half a block before two women ran over to snag a selfie. A man shouted out his name from a bar running on a generator and offered to buy him a rum sour.

The reaction is more subdued in rural mountain communities like Naguabo, where Mr. Andrés and his crew have been delivering supplies so cooks at a small Pentecostal church can make 5,000 servings of arroz con pollo and carne guisada every day. There, people touch his sleeve and whisper, “Gracias.” They surround him and pray.

“He’s much more than a hero,” said Jesus R. Rivera, who was inside a cigar store watching Mr. Andrés pick out one of his daily smokes. “The situation is that still some people don’t even have food. He is all that is keeping them from starving.”