Lt Col Gray's hat does not have the gold braid called "scrambled eggs" on the visor. As a Lt Col his hat visor would have that.
Even in 1967, no one could have driven to such widely-flung places around Southern California, in the given time-frame, as Gomer did. He had seven hours to get to Wilshire Blvd in downtown LA, but first drove to and, in some cases, spent considerable time in: Van Nuys (in the San Fernando Valley), Pasadena (in the San Gabriel Valley), Santa Monica (on the coast), and Disneyland (in Orange County).
The staff car that Sergeant Carter & Colonel Gray uses to find Gomer Pyle is the exact same staff car that Gomer has been using to do his rounds in the city. Both have serial number 221017.
The 1967 Dodge Coronet staff car Gomer drives throughout the episode, and the one Sergeant Carter and Colonel Gray later use to search for Gomer, both have the same motor pool serial number on the doors. Since the two cars never appear together in the same shot, it's a safe bet the director used the same car to simulate two different cars.
Colonel Gray briefed Sergeant Carter that General Prescott is to be picked up by Gomer Pyle at the Chancellor Hotel on Wilshire Blvd. When Gomer Pyle finally arrives, the hotel is shown to be past Berendo Street and the highway in which the hotel sits on ends straight ahead. Wilshire Boulevard does not end at either side of Berendo Street.