I have installed and activated the plugin "Site Maintenance Page". Works perfectly. However, if you log out and try to log in again, it does not work, the Maintenance Page always appears. Even if I enter mysitename/admin as the address.

Edit [plugin directory]/controller/SiteMaintenancePage.php
and replace the whole WhichPage function (line 23 downwards) with this one:

    function WhichPage($page){
        global $addonPathData, $addonPathCode;
        $type = \gp\tool::SpecialOrAdmin($page);
        if (!common::LoggedIn() && $type != 'admin') {
            if ($this->config['enabled']) {
                global $smp_config;
                $smp_config = $this->config;
                gpPlugin::incl('view/maintenancemode.php');
            }
        }elseif (common::LoggedIn() && $this->config['enabled']) {
            msg('NOTICE: Site Maintenance Page is enabled. '
                . 'Only logged in users can see the site. ('
                . common::Link('Admin_SiteMaintenancePage', 'Disable Maintenance Page now', 'cmd=disable')
                .')');
        }
        return $page;
    }

This should do the trick.

... oh and then use [your website url]/Admin to log in

edit: I crossposted it to the Plugin Support Forum
edit2: Please report back if it actually works reliably - I only tested it quickly.

Hello Jürgen,

now it works like clockwork. Thumbs up!