Server Migration Solution???

We are trying to make a migration from one dedicated server to another but we are concern that if may have any problem after the dns nameservers will change because even that all files can be transferred our website platform have some custom configuration which may cause problems at the new server.Our request consist in how to test our NEW SERVER which access the same files like the OLD SERVER and both of them to be online.A logical solution is that Old Server is related with Old nameservers and the New server to be related with the New Nameservers but when a global user tries to access our site expample.com we want them to show the OLD one SERVER in the other hand we want to find a way that we as the administrator can access the same domain name "example.com" to test the platform from the NEW SERVER.Is there a way how to do this ? Thank you.