A warm compressor actually is a good thing. This ensures there is no condensed refrigerant in the compressor. The amount of heat stored in the shell is not huge. Having a warm compressor on start-up actually helps the condenser hx start doing its job earlier, shortening capacity ramp-up from equilibrium.
The best easy thing you can do after insulating the box is to insulate the suction line. I use armaflex closed cell foam pipe wrap. The heat not gained by the exposed piping helps the compressor run cooler by reducing superheat.
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