A different approach:
If your Strat clone won't stay in tune, and if it has a "whammy bar" you might be able to fix the problem with a "hard tail" Strat bridge from GFS:
http://store.guitarfetish.com/Chrome-Hardtail-Bridge-for-Teles-and-Strats_p_158.htm
$11.00 is a lot less than another guitar.
Another problem could be the tuners -- although I'd try the bridge first. Again, at GFS:
Nickel Vintage Fender-Style Machines/Press fit Bushings
Even if you end up buying the bridge AND the tuners, you will be well under your $235.00 budget figure.
RE: Tone. You can't really count the cost of strings -- you'll have to buy strings now and then even if you buy an expensive guitar. If you haven't tried this already, buy a set of strings explicitly targeting jazz -- a medium gauge set of D'Addario chrome flat wounds for example. See what the Strat clone you own already sounds like with just the neck pickup and these strings.
Almost any guitar that sells new for less than $1,000 can benefit from a pickup upgrade and you might eventually want to reward yourself with a new pickup (although I would put that off until the money situation improves). A $99 Strat knockoff is something you probably wouldn't mind attacking with a router and a chisel if you can borrow the tools, and something like the Lollar CC pickup in the neck position sounds great to my ears after enough wood and pickguard is hogged out to allow installation.
Obviously, that pickup is over budget, but there is always next year, and if you can take care of the tuning issues now for $11.00~$145.00 this year with the bridge and maybe tuners -- you can play the hell of out your Strat clone for a year while you save up a little more money for a pickup as a reward to yourself for being frugal.