The latest term means it’ll have the highest term_ID right, So we’re sorting terms in DESC order and getting first array element of the result in $latest_mailer variable.
Here’s code to do that, not tested but if didn’t worked – you can modify get_terms() function to return term with higest id.
$mailer = get_terms('mailers','orderby=id&order=DESC&number=1');
$latest_mailer = $mailer[0]->slug;