Switching power to an air compressor
I would just use a standard air conditioning condenser relay rated for 30 amps or more to start with. Make sure it is rated for horsepower larger than your compressor. (To really calculate the inrush current you need the code letter from the motor nameplate under the LRA or locked rotor current. Then see Table 430.7(B) in the NEC. Multiply your horsepower by the number from the table and divide by one thousand. See this article here.)

Or just start with an A/C relay like I said above and if it wears out fairly quickly you could spend much more money for a higher rated motor contactor. An A/C relay probably costs $30 or less.

The other thing to consider is when you decide to upgrade your compressor, make sure it is a two-stage compressor. A two stage starts the motor unloaded and then engage the compressor after the motor is running. This significantly reduces the inrush current to the motor since it is starting unloaded. The single stage style is required to start with the compressor under pressure and consequently requires a large inrush current.

Good luck and stay safe!