I actually had a very similar problem with my S-W horn and it came down to the fact that the S-W and Sparton horns use different adjustment screw setups. The S-W screw is longer, and the bridge for it has the serrations cut into the bridge, while the Sparton screw is shorter and uses a smooth bridge, with serrations cut into a clip that goes over the bridge.
Most reproduction parts are based on the Sparton horn, so if yours was restored using Sparton parts, that could explain the problem. You might try to find a donor S-W horn to steal the adjustment screw assembly from; however, these tend to be the first parts to go since they are exposed to the outside.
I ultimately solved the problem on my horn by using the Sparton bridge setup with the original S-W screw. It might also work the other way around, I don't know. Using the longer screw allowed me to get the proper pressure on the ratchet to produce the right sound.