The retiring chief of naval operations, Adm. Carlisle A.H. Trost, is scrambling to find a good job for his friend, a two-star admiral who has presided over a series of scandals at the Naval Academy in Annapolis.

Trost wants to pin a third star on Rear Adm. Virgil L. Hill, the academy superintendent, and find Hill a job befitting a three-star admiral before Trost retires. The retirement ceremony is set for June 29 and our sources say Trost's plan is to sign new orders and a promotion for Hill on June 28 or 29. But some top Navy officials are in an uproar over this bit of lame-duck back scratching.

The academy is a prestigious institution that has graduated luminaries of the Navy, public service and industry. But during Hill's two-year tenure as superintendent, the academy has been rocked with scandals and is now the subject of at least six different military and congressional investigations.

Hill's tenure at the academy has been a minefield, and our sources say not all the mines have been detonated yet. There have been allegations of unchecked cheating, uncontrolled hazing and a watering down of academic excellence. Knowledgeable sources told us another major scandal is about to break regarding the mistreatment of Hispanic midshipmen.

By now Hill should have his scandal response perfected. One of the first occurred in the 1988-89 school year when a spoof magazine published by middies, called "Playmid," printed photos of male and female midshipmen in scanty garb.

Hill overreacted to the joke, confiscating the magazines. But someone broke into the storage room and stole some back. Hill called in the heavy-handed Naval Investigative Service. The final straw was the publication in Playboy magazine of the photos. Hill had managed to escalate a case of sophomoric hijinks into a national embarrassment for the academy.

Then the scandals grew more serious. One of Trost's assignments for Hill was to make the academy more oriented toward high-tech studies, but Hill has given the wrong signals. For instance, he removed a respected teacher, Ralph Santoro, as chairman of the electrical engineering department after Santoro refused to raise some grades.

This spring, Hill deliberately looked the other way when there was evidence that midshipmen copied engineering exams during a break-in of faculty offices.

The school year ended with another scandal when midshipman Gwen Dreyer resigned, saying that upperclassmen had handcuffed her to a urinal and taken pictures. Another midshipman, Victor Vaca, resigned after he said he was forced by classmates to eat and drink until he vomited.

As if the bad press were not enough, the 990 graduating midshipmen this year were given diplomas which misspelled "Naval" as "Navel."

Just where Hill will go is in Trost's hands. Technically, Trost could promote Hill to a three-star vice admiral and keep him at the helm of the academy. But our sources say Trost wants a more important command for his friend, maybe Atlantic naval operations.

Footnote: A spokesman for Trost vehemently denies he is trying to promote Hill or reassign him before Trost retires.