A fridge is a metal box and it is very difficult if not impossible to get radio signals in or out. So you will need to mount the module that sends the SMS outside the fridge. Ideally, only the temp sensor will be inside.
You could also email yourself an SMS, if your phone service provider allows you to do so.
For example, I can email myself an SMS by sending an email to something like myUSphoneNumber@sms.cricketwireless.net
Each carrier has a different @xxxxxx.yyy
Even if it is possible for you, I'm not sure it has any advantage (vs just sending an email to yourself).
I recommend an ESP8266 for email (or ESP32) with the mobizt ESP-Mail-Client library. No need for a Nano or other Arduino. The ESPs are perfectly fine stand-alone devices. And they're inexpensive.