According to your process, you are technically correct. +966 is the country code in Saudi Arabia and must be used for International Direct Dial. It may be that the phone number you get is really invalid. Please double check it with your family.
I recommend you to use internet calling / WiFi calling in IDD since IDD is REALLY expensive (unless money doesn't matter to you). Some of them are Viber, LINE, Kakaotalk that only use your data when calling.
+966***** . Have you also tried this?