Side By Side Upgrade and Migration from 2008 to 2016
I've had to do this and just had a record available for the old server.  Renamed the DNS entries and did a ipconfig /flushdns on the servers themselves since they were replicating back in case of a failback.  We set the DNS to a very low TTL so clients also got updated.  Powershell made all this very easy and fast.

If your clients are easily identified and you have access, you can do a flushdns yourself on them as well.  I recorded and documented all connections going to the database prior to the upgrade so I knew all the clients to look out for.  Then finally I set the old database to block all users except the sole one I was using for replication on the old machine and ran traces to see if anyone was attempting to connect to the old instance.  None did but if they were, I would inform them to update DNS or check their connection string.