The underConstruction(...) plugin is useful if you have a test site that you don’t want people to access unless they’re authorised to it.
I’ve installed it on www.gordontate.org.uk
Anyone who accesses this site will be gently prompted to click on a button to connect to www.gordontate.co.uk

Unfortunately, this plugin doesn’t appear to work for WordPress MultiSite websites which are created as subdirectories of the main site. e.g. www.rowlandscastlewebdesign.com/rcgc. When you configure it on WPMS it applies to the main domain. So, until I fix it, trying to connect to www.rowlandscastlewebdesign.com shows the “under construction” page intended for the mythical golf club called “Bob bling Wide golf club”

So I’m going to try Members Only(...)