You're looking for a "Template Override". To do that, copy the /modules/mod_sclogin/tmpl/logout.php file to /templates/<YOUR_TEMPLATE>/html/mod_sclogin/logout.php . Then, edit the file there so that updates to the module don't override your customizations to the layout.
To do what you're looking for, I think you should just need to copy the following line:
echo '<div class="sclogin-greeting">' . JText::sprintf('MOD_SCLOGIN_WELCOME', $name) . '</div>';
To the section below:
<?php
}
if ($params->get('showUserMenu'))
Like:
<?php
}
echo '<div class="sclogin-greeting">' . JText::sprintf('MOD_SCLOGIN_WELCOME', $name) . '</div>';
if ($params->get('showUserMenu'))
Of course, test and make sure that does what you're looking for. If you need more assistance though, just let me know.
Thanks,
Alex