So ...

We are going to use a handy plugin called WP Maintenance Mode.

Here it is:

https://wordpress.org/plugins/wp-maintenance-mode/

It's a very big plugin (5.46 MB), so let's see the usual speed tests ...

Here are my results before installing the plugin ...

... and the result after installing and activating the plugin:

As you can see, it's a big plugin but is definitely a well-coded one. No more comments needed ...

Now let's see how it works ...