Domain Mapping Setup Help

I have a new multisite install (subdomains) that I want to setup domain mapping on.

The IP address I have the multisite install on is also being used for another multisite install with domain mapping already configured. So when I test to see where the IP address is directed, it points to the original site. Not my new one.

I’d prefer not to request more IP’s for the new site.

Can someone please explain another way to set this up so that I can get the domain mapping for the new site to also work.

Thanks for any advice.