Using is_user_logged_in() to lock down whole site

Just before any of the frontend template loads, WordPress fires an action – template_redirect. You can use this action hook to show SAML login if user isn’t logged in. This way, you won’t have to put the condition in every template file.

// you can place this code in your theme's functions.php file.

add_action( 'template_redirect', 'wpse_login_required' );

function wpse_login_required() {
    if ( ! is_user_logged_in() ) {
        // a link via SAML to login to the site
        // ....

        // kill the execution
        exit;
    }
}

This way, the content will never be public as well.