Noob Question – Site Backup

Hello! Sorry if this is a noob question, I’ve been searching and searching but can’t find anything that’s EXACTLY the same idea, and I do not want to mess up my current site.

Domain: Oldworldlabs.com

Here’s my dilemma. I uploaded the old site database into phpmyadmin and my old site came back. Great! The problem is my new (actual) disappeared. I have backups for that too, if needed. What I want to do is have my old site running on oldworldlabs.com, my main domain, while having my new website that I am still working on in another directory or even sub domain. For example, oldworldlabs.com/new. Whichever is easier! It may be really simple, I just can’t find the right answer currently. Thanks for reading!