Moving a website to a new host while keeping the current site alive.

I have developed a website for a client under a dev subdomain on my server.

The client currently has an active site at Hostgator.

The client purchased hosting space at A Small Orange.
The space is registered under
His domain is managed by a third party through GoDaddy.

I am finished with his NEW site.
I have been trying to move the NEW site to the NEW hosting at A Small Orange.
BackupBuddy failed.
Duplicator failed.
My manual efforts failed.
It is hard trying to troubleshoot because the urls for troubleshooting keep reverting back to his EXISTING site.
I have changed my hosts file, but it works spottily.

Can you offer any suggestions/best practices on relocating a dev site while an old site is still active on another host?

Thank you.
Brenda Malone