hook_mail_alter() - how to access the "contact admin" form fields?
Try doing this in hook_form_alter() instead of hook_mail_alter(). You can likely then run your own parameters into drupal_mail().

I would recommend not using this contact form, though, since it's terribly limited in terms of functionality. Why not use a Webform at /contact where you can specify all of these settings and do potentially cooler things too, like pre-filling fields for logged in users?