Old Site Live while New Site Under Dev – DNS Switch vs. Redirect

I’m fairly new to this process having built only a few WP sites. In each these cases, there was an existing site that needed to be kept live until the new WP site was ready to go. I’ve tried a variety of ways to manage this effort, but none of my approaches have been without issues. Looking for an ideal approach.

In all cases, we’ve also moved to new hosting server with the new site (while maintaining the same domain), so there are options with when we switch the DNS.

A few things:
I’d rather not port the old site (non WP) to the new server.

I’d rather not update my local hosts file to point to the new IP to do the development. Everyone else who needs who need to see the site during development would need to do the same, and I’d rather not ask clients to do this.

I’d love to switch the DNS and put in place some sort of redirect from the under development site to the old site, but didn’t know how to this without creating some sort of loop since the domain name will remain constant. Is there a way to redirect with an IP reference to allow for this?

I’m sure there is some great way to handle this as it’s far from an unusual situation. However, I have not been able to find one.

Any advice would be much appreciated! Thanks!