Change the current loggedin Domain?
It depends on your settings.php $cookie_domain value.

/**
 * Drupal automatically generates a unique session cookie name for each site
 * based on on its full domain name. If you have multiple domains pointing at
 * the same Drupal site, you can either redirect them all to a single domain
 * (see comment in .htaccess), or uncomment the line below and specify their
 * shared base domain. Doing so assures that users remain logged in as they
 * cross between your various domains.
 */
# $cookie_domain = 'example.com';


For the session cookie, a value of:

$cookie_domain = '.example.com'; // Note the leading wildcard dot "."


will work across subdomains

Whereas a value of:

$cookie_domain = 'true.example.com';


will only work on true.example.com

If that session id is valid for that site database session table then the user will be logged in.