Replacing the capacitor with something like a .1uf will make for a more drastic shift in tone control. The "rolled back" sound will be much more muffled, like dub/motown style bass. When wide open, it's still clear.
And yeah, it's possible that the pot has kinda gone bad, so if you were going to that level of effort, I'd say replacing the whole tone pot would be a good move.
Also, with that in mind, you could get a tone pot that has a push/pull switch and do something like a "dual cap" version so you can have the traditional .047uf and the .1uf available, all you have to do is set it in the mode you like. It's not much more than a basic replacement either.
I do them in a lot of basses and no one who's tried it has disliked it.
And yeah, it's possible that the pot has kinda gone bad, so if you were going to that level of effort, I'd say replacing the whole tone pot would be a good move.
Also, with that in mind, you could get a tone pot that has a push/pull switch and do something like a "dual cap" version so you can have the traditional .047uf and the .1uf available, all you have to do is set it in the mode you like. It's not much more than a basic replacement either.
I do them in a lot of basses and no one who's tried it has disliked it.