A trip to the archives clears up the mystery about what became of the Hippodrome. As I suspected, the back of the theater was demolished, but the facade was saved.
Notice that the Main Street Gym doesn't seem to have set up business yet (the sign over the door reads "Here Television," although a 1984 article says that the gym moved to 318 1/2 S. Main in 1951 after its old building across the street burned down). There's a cafe at 318 S. Main, which was replaced with Mexico Luggage and a barbershop by 1984.
Notice that the building was also called the Adolphus Theatre. This is another mystery. According to The Times, the Adolphus opened on Main Street "across from the Belasco" (335 S. Main St). on Nov. 27, 1911. It vanished from the clips after July 14, 1912. The Hippodrome opened Aug. 31, 1913. A one-year gap would allow for construction of another theater (the Hippodrome seated 3,000 while the Adolphus seated 1,450), but I can't find anything in the clips. Stay tuned...
And here's the demolition of the Hippodrome's proscenium, Oct. 26, 1952.