FRESNO, Calif. — Millennials just weren’t eating raisins. So Sun-Maid, the century-old company with the iconic little red raisin boxes, hired someone to convince them that they should.

At 38, Harry Overly was decades younger than the tenured raisin man he replaced as the chief executive of Sun-Maid. But he had experience — as the North American head of the company that makes Bertolli olive oil, and in marketing roles at Wrigley and other food companies. He seemed suited to the job.

When he came west, though, he was taken aback by the level of animosity he encountered in the U.S. raisin industry, the entirety of which is crammed into a few hundred square miles in California’s Central Valley.

Three months into his tenure, which began on Halloween of 2017, Mr. Overly attended a meeting of some raisin industry players in the back room of a restaurant in Fresno, Calif. This introduction left him shaken. “I’m not saying this lightly, because — you can read about this in different spots — people kind of think there’s this raisin mafia out there and that kind of stuff,” Mr. Overly said.