I've done some research on the Mighty Mini, and they claim it can pump out 30psi which is enough for any airbrush, so you might get by for awhile without upgrading the compressor. I would suggest getting a regulator for it to keep the pressure between 12-20psi. A $10 regulator from Wally-World or Harbor Freight would do (it's what I use).
If you have the A270 AB that comes with the kit, I can see why you're having trouble! The 250 isn't a whole lot better, and is more a craft brush than a modelers brush. The Paasche H or Badger 350 is a better siphon fed external mix than the all plastic A270 Aztek, but I would go for broke and get the 200 or 100LG.
If I were you, I would upgrade the airbrush first, but start thinking about what you can do about the compressor. Even putting $5-10 under the mattress will quickly add up to a pretty decent compressor in a very short while. The compressor I've linked to is a fairly decent compressor.
Here's a thought! Badger still has the 200NH siphon feed for $22 and the 100G for $30. The 1/16 oz cup on the 100G would mean refilling more than once to paint a car body, but that's not that onerous. $30 for the 100G is a smoking deal, and either brush would be a huge step up in quality from where you are at right now. Perhaps you could justify getting the compressor at the same time? If I believe Harbor Freight shipping rates, it's only $9 to ship in the US! You could pickup a vinyl hose at Michael's for around $10. You would end up with a pretty decent setup for $100.