I have only done this once and I am at a different server (BlueHost), but I have two sites that are virtually identical and yet completely separate...and I think that is what you are saying you want. After the first site was working (like you have now), I installed the second site as a sub-domain (by using SimpleScripts at the BlueHost Dashboard) into its own new folder inside my exiting public_html folder. Then when the SimpleScripts action had completed, it sent me an admin login for the second site and all was good-to-go.
Hope that helps.