While it could be the clutch cycling switch... you need to remember the pressure temperature relationship of refrigerant. The warmer the ambient temperature... the higher the pressure of the refrigerant in your a/c system.
The clutch cycling switch won't allow the compressor clutch to engage if system pressure is below the upper threshold. That is typically 40 to 45 psi. Then the switch will cycle the compressor clutch off when the system pressure falls below the lower threshold... typically 20 to 25 psi.
So when you start the engine and the compressor clutch won't engage... it could be the system is low and doesn't have enough pressure. As the engine warms up... ambient temps underhood increase... which in turn raises a/c system pressure. If it achieves pressures greater then the upper threshold... the clutch engages.
If that is what is going on in your case... the clutch will cycle on and off fairly quick. The performance of the system will also be degraded and not blow acceptable levels of cold air.
If you try a low pressure switch and it doesn't change your issue... you'll need to put some gauges on the system to further diagnose the condition.
Hope that helps... and good luck.