Do I need to use drupal_mail after the hook_form_contact_site_form_alter?
You should use hook_mail_alter().

hook_mail_alter() allows you to modify the email cleanly before it is sent. The Contact module passes the entire form state to drupal_mail(), so you have access to it in this alter hook via the params key.

A quick example. I haven't tested it:

function my_module_mail_alter(&amp;$message) {
  if ($message['id'] == 'contact_page_mail') {
    $phone = $message['params']['phone'];

    $message['body'][] = "Phone number: " . check_plain($phone);
  }
}


Let me know if that works.