Assuming you're on the OEM Michelin Summer tires, the biggest limiting factor is the heavy OEM wheels. The front 225s are well supported on 8" wheels. Going to 235s won't net you much gains in front grip in an AutoX setting. And, unless you also increase rear tire rolling diameter similarly, just going wider up front will increase the front ride height and you'll end up with a reverse rake and worsen Stinger's understeer tendency.
The rear 255s are already the widest 8.5" wheels can support well, so going wider in the back won't help much either. Besides, even if it does, given Stinger's tendency to understeer, better grip out back is just going to worsen that.
That said, if your stock tires are all season, then that will be the weak point, especially in the hot Summer season. Since you're in SoCal, you'd want to switching to as good of a Summer tire as you are willing to tolerate the rapid tire wear for street use.
Realistically, if you are just doing an occasional AutoX, go out there and have fun, and don't worry about too many things. IF you are serious about SCCA Solo, then starting with a dedicated set of lightweight wheels is you best bet. Regardless of what you do, concentrate on improving the driver - not the car. That's what it's all about. Stingers are far from the best AutoX cars, regardless of trim, and won't be much competitive, but plenty good enough of a chassis to learn on.