New website under existing website; how to switch

Hey everyone, need some help.

I have a live site with woocommerce on a domain.

I wanted to redesign the page, and since i didnt want to take it down in the process, I started building a redisgned site in a folder in the root of the site and created a new installation there.

So:

existing site is on http://www.example.com
new site is on http://www.example.com/newsite

So i have my current site working on my domain, and the new site is finished and ready to go live on the domain.

How should i go about making the new site live?

Thank you in advance.