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The cap sets the corner frequency of the tone control. Small value for treble, larger value for treble and high mids.
Tolerance/ accuracy of pots is described as a percentage. The steepness of the log curve of a log pot is also expressed as a percentage - be careful you know which detail you are looking at.
Lots of euphemistic terms in guitar cicles for the different %log tapers. I recall that either @gavin_axecaster or @sixstringsupplies once described who sold what versus actual %tapers but can't find the comment just now.
So, log pots come with different curves, typically between a 10% taper for a modern pot up to 30% taper. A linear pot is at 50% resistance at 50% rotation and reaches that via a predictable straight line (give or take), eg 250K for a 500K pot. By comparison a 10% log pot would be at 10% of its resistance at 50% rotation, achieved via a curve, eg 50K for a 500K pot. Clearly a 10% log pot rolls off treble very much more quickly than a linear pot. And a 30% taper pot would sit between 10% log and linear.
If you don't like the mud in the lower reaches of the pots range a fixed resistor in the path to earth will effectively ensure the pot can't get to 0K. For example, if you never use the tone pot below '3' (on a knob scaled 1-10), and '3' represented 5K resistance, then a 5K fixed resistor means your tone pot is now at '3' even when turned fully down - you've now got greater resolution of control for the tones you actually use as the '10' down to '3' range has been extended '10' down to '1'. The 5K resistance will have negligible effect at '10' as its such a small percentage greater relative to the 500K of resistance of the pot. This trick is part of Fenders grease bucket circuit, which I mostly don't like, but the fixed resistor aspect of that circuit (as described above) can be useful.
http://www.thefretboard.co.uk/discussion/89942/caspercaster#latest
Failing that, a 250K - if it’s currently a 500 - would possibly help, although the tone at 10 will then be slightly duller. I’m not actually a fan of having the tone pot value lower than the volume at all, to me it never sounds quite right.
"Take these three items, some WD-40, a vise grip, and a roll of duct tape. Any man worth his salt can fix almost any problem with this stuff alone." - Walt Kowalski
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I wish I could remember if it was audio or linear, I used to know everything about how I wired my guitars all up off by heart but after a serious medical incident I lost a bit of memory so now have to try and work it out from diagrams. Think I'll have to just open it up and check though, I think I remember that it says A or B on the actual pot. The 250k thing sounds interesting cause the pickups are fairly bright for humbuckers so having them slightly dulled at 10 might not be so terrible.
It's a CTS one so not particularly cheap. It's the middle terminal that's connected to the volume pot's input. Does that have an effect on how it works I take it?
Some (most?) of the pots say on the back what the value and taper are (though if you've soldered over that, that might not help!). Alphas say on the top as well (I don't think CTS does, but I could be wrong) but you have to take them out to see that.