The police hat was the common headwear for police officers before the War. It can usually be found near a skeleton's head with some form of low caliber pistol. It provides a Damage Resistance of 1.
In the presidential metro, after you exit the hall with a few flags and the statues of presidents immediately turn right, it is on the bench there.
By a skeleton on a bench near some of the tracks. To find this is easy, after you exit M.A.R.Go.T.'s control room you will see an American Flag, go that way and turn left towards the tracks, while facing the tracks make another left. Go towards the caved in part of the tracks, and near the wall you will see a skeleton with a .32 pistol and a police hat.
A second one is located in the center of the Old Olney S. Wilson Building. It is accessed by jumping from a ledge adjacent to the fallen GNR billboard onto the empty deathclaw cage. The police hat is next to a skeleton on a small platform at eye level.
A police hat can also be found on top of a television in the Grower's Shack.
One is located inside the lighthouse halfway up the stairs in Point Lookout.