One of the challenges of consolidation and upgrades is that some applications might have hard coded or difficult to find references to the server instance they are connecting to.
When you want to point this hypothetical application to the non-default instance on a new server you can often use DNS to create an alia to a specific server \instance combination providing the alias (i.e. the legacy server) is no longer around.
However for side by side upgrade in a non virtual world, another simple tip is to fiddle with the SQL Server connection manager:
Now I can connect to the named instance by simply typing host server name (or “.”) through SSMS or the legacy application.
I wouldn’t recommend this approach for any internet facing server as you will probably not want to use the default port of 1433 at all.