The coach of the Mount Pilot Comets team is named Mr. Carter. The subtitles on Amazon Prime are incorrect. They have Andy calling him "Mr. Tyler."
An almost identical plot occurred in The National Pastime (1961), in which Steven Douglas was called in to umpire his son Chip's baseball game and called him out at the plate.
For some reason there are no numbers on the teams uniforms.
When Andy and Helen are walking after the movie, the building fronts they pass by were used earlier for Mrs. Luken's woman's-wear in "The Bed Jacket," followed by a jewelry store, in the early black-and-white seasons. The building that Andy is standing in front of when the little dog barks at him was used for William Christopher's office in "A New Doctor in Town."