From what I remember, there are large schedule boards on each of the platforms at La Spezia Centrale. When I was there last fall, I checked the schedule quickly, and asked a station agent (who happened to be on the Platform) to confirm which train to Board.
The C.T. train wasn't hard to spot, as it's a bit older than the IC or ICE trains.
There's undoubtedly several ways one can get to the C.T. I certainly agree that it's not desirable to have to go down through the "Sottopassagio" and then up to the ticket window, as it wastes a lot of time.