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FWIW: I tried a set ot Mickey Thompson ET Street Radial II tires on the street. I ran them for a total of 70 miles to and from work one day to see if they were a viable option. They weren't! Even inflated to 38lbs they were unstable and absolutely killed the handling which is a premium to me. They did provide a lot more traction but I considered them dangerous for my style of driving. I took them off and now run L5 rear quarters with 345x30 PS2's. Great ride and handling and "almost" enough traction for driving quickly. I not the type that tries to light up the tires at every stop sign and light but I do seriously enjoy the country roads where I live.
The real solution to you problem may be a driver mod.
OK, please no more comments from the STAR people.
My original post asked for,
How much biigger can I go an original rims,
No DR"s
What is the flavor of the day ...for sticky steet tires...
In very general terms, the lower the UTQG Treadwear number the quicker the wear due to having a softer the tread compound. And, a softer tread compound generally means more traction. So, your NT555 tires are rated at 300. Try a good quality tire rated < 200 and see if it helps. Take a look at tires like the Bridgestone Potenza RE-11 or the Yokohama ADVAN Neova AD08.
I would expect you can go about 1 size wider on the same rims. Look at the tires and the recommended rim size and generally you want the actual rims to be about in the middle of the recommended range.
Lastly, all tires will get "harder" as they get older and have miles put on them. So, your new sticky tires after being used for 4 years and 20k miles just won't be very sticky anymore.