I think there is no point in hunting down the incoming SMS as they are available in the GSM module.

You can once your MCU wants or is available to ask for the new messages, and select the most recent or decide from the list which one you want to read, deleted etc. Your GSM module may receive also messages from the Network operator, messages by error from someone that typed wrong number or ads. So you better check the list and see who sent it, then process it.

In this case you do not need Interrupts. Maybe if you want to raise a flag that a new SMS has arrived, but anyway, it is not necessary.

Also On Interrupts are slow by default! I think I never used them.

Ioannis