"Mayberry Goes Hollywood" originally aired January 2, 1961. In the jail cell that Andy and Barney clean hang two calendars (with months and years obscured as usual). One calendar features a month of thirty days with the 1st on a Tuesday, most likely making it November, 1960. Over Andy's shoulder, another calendar hangs. It's only partially seen with the 28th appearing on a Saturday. The closest months to the air date when this would be true would be either November, 1959 or April, 1961.
When Barney comes out of the back room wearing his new uniform, his hat is on incorrectly. The lanyards on his hat should be facing forward, not 90 degrees off to the left.