Logic thinking to solve a problem that occurred on unfamiliar system

Logic thinking to solve a problem that occurred on unfamiliar system. I have a chance to migrate a customer’s firewall in Brisbane, Australia. The customer also need to change his service provider at the same time. To change their service provider, it means their public ip address will be changed as well. This task was belong to the third-party which we had to work with them, to wait them finish their job and then we go on migrate the firewall. After, we finished with the migration. We tested the system. Did the VPN over there working? Can they travel through the VPN tunnel to connect their HQ? Was the tunnel stable enough? Did the Failover working? Did their DNS also functional? And so on. Everything were seems alright but there was one problem exist. It was user’s E-mail system. Please share me your way of thinking to solve this problem. I have presentiment that it was certainly the changed public IP address. By the way, w...