As a college student at New York University, Morgan Riddle paid almost no attention to tennis. But she always knew when the U.S. Open was underway at the end of August.

“When I was out in the city for my internships, I would see the hats,” Ms. Riddle said.

Ms. Riddle, 27, an influencer and the girlfriend of Taylor Fritz, the top-ranked American in men’s tennis, is now the owner of two different examples of the tournament’s signature piece of merchandise: a basic ball cap featuring the year and a flaming-tennis-ball logo.

Each year, thousands of U.S. Open hats leave the Billie Jean King National Tennis Center in Queens on fans’ sweaty heads. By early September, the hats trickle into the city’s gyms and onto its pickleball courts.

Ms. Riddle has a blue one and a white one in her vast closet of tennis apparel. A camouflage version has caught her eye, but she is not yet convinced: “I don’t know how I feel about it,” she said, laughing.

Each hat costs $40, a steep price for fans who may also be balancing the cost of tickets and the tournament’s other marquee accessory, the $23 Honey Deuce cocktail. But many consider the bragging rights worth it: More than 200,000 U.S. Open hats are expected to be sold this year, said Mary Ryan, the senior director of merchandise and licensing for the United States Tennis Association.