The simplest solution is to use HTTP Basic Auth, which would put the whole site behind a username/password. This won't interfere with laravel at all, and is completely outside of laravels auth. In other words, you browse to the domain and immediately see a username/password prompt before you can see any content. Once it's entered, you'd then be allowed to browse the site and use it normally.
It's pretty easy to set up, but it differs depending on the web server (nginx/apache/etc). You can even set up multiple users/passwords and see which ones are accessing the site in the server logs.
I'd do that, and then just remove it after it no longer needs to be private.
For Nginx: https://docs.nginx.com/nginx/admin-guide/security-controls/configuring-http-basic-authentication/
Another possibility is to allow access to the server by specific ips, and deny everything else. That's more problematic, though.