If it's wired like the only single pickup, single volume control guitar that I've ever opened up (that would be the kid's Daisy Rock), then there's no tone cap in the circuit. If you wanted to keep the same look you have, you could use a concentric pot, with volume and tone on the same pot (stacked). That way you could add a tone cap that suits you, as well as having [some] control over your tone. All the above suggestions are good ones, btw.

If it were me, I'd get an AlNiCo humbucker to replace the stock pickup, and probably add a tone control. Also, it would take me four years to complete the project.