I believe After the migration is complete, you can update the DNS records to point the existing FQDN (e.g., xxxxx.postgres.database.azure.com) to the new Flexible Server's IP address.
Or If your Flexible Server is behind a custom DNS server or uses custom DNS namespaces, you can set up a DNS entry to resolve the new Flexible Server to the same FQDN that was previously used by the Single Server
Here is related information from the migration document https://learn.microsoft.com/en-us/azure/postgresql/migrate/migration-service/tutorial-migration-service-single-to-flexible?tabs=portal%2Coffline#select-migration-target
Select migration target
The Select migration target section displays metadata for the Flexible Server target, such as Subscription, Resource group, Server name, Location, and PostgreSQL version.
The Custom FQDN/IP field is optional and can be used when the target is behind a custom DNS server or has custom DNS namespaces, making it accessible only via specific FQDNs or IP addresses. For example, this could include entries like flexibleserver.example.com, 198.1.0.2, or a PostgreSQL FQDN such as flexibleserver.postgres.database.azure.com, if the custom DNS server contains the DNS zone postgres.database.azure.com or forwards queries for this zone to 168.63.129.16, where the FQDN is resolved in the Azure public or private DNS zone.
Thank You