What used to be, and what currently is may be completely different these days, but you used to be able to replace the solenoid on the old GM starters. You have to remove the starter from the car to replace the solenoid.
That said, I have to wonder if since someone ran a wire down to the starter, if the problem lays elsewhere (unless the starter is a royal pain to change). I'm betting the extra wire you spoke of bypasses everything between the battery and the starter. Someone didn't know how to chase down the problem, or couldn't find the problem, so they just added the extra wire. It is possible that simply replacing the extra wire with another will make it work "like it always has".

Personally, if it was mine, I'd replace the starter as a starting point, if its still the original, its spent a lot of time just sitting, or it could be completely wore out. You will have to pull it out to do just about anything with it, I probably would not put that old starter back in without a complete rebuild unless you know its full history. We have places in our town that actually rebuild the old starters and alternators, might be better then the new Chiniseum stuff on the market these days.