WASHINGTON—At a gala dinner in February to raise funds for the families of Central Intelligence Agency officers killed in the line of duty, CIA Director Gina Haspel surprised her audience by delving into details of spycraft the agency has used to run agents on the streets of Moscow. But the crowd’s astonishment at the unusual revelation quickly evaporated when the spy chief confided that the material came from a journalist’s book.

After a year atop the CIA, Ms. Haspel is giving away few secrets. With a 35-year career in clandestine...