I'm guessing there's a gain in entropy inside the tire as well. Increasing the pressure is equivalent to increasing heat. On the other hand, the maximum number of positions occupiable by gas molecules has to decrease, but you aso have more gas molecules.
I was kind of wondering about the heat issue as well, and I really don't know what to make of it. The molecules are more restricted in their range of motion, but they move faster. I don't know whether that counts as an increase, a decrease, or neutrality.