Backup copies movement and retention

We are using Data Domain as backup destination for servers in a bunch of server groups.
When we run a second backup copy of the same bunch, we want first copy to be moved to another location within the same Data Domain,
as our backup tool can't overwrite the existing backup with the same name.
So, we want the previous copy to be moved to another location within the same Data Domain. That way we want to retain all the previous copies. How this can be accomplished in Data Domain. Snapshot/FastCopy ?. And, how the copy part on the DD can be scripted.
Have anyone done this. Is there any similar use case available with anyone. Please advise.

Thanks