Modules/Methods for Faster Data Entry
This particular solution was a custom form, where I had fine control over the CSS classes applied to the different elements, but it might also work for node-edit et al.

You would use hook_form_alter() to target whichever form you want.  Then, you would add a custom CSS class and an '#attached' property to that form element, containing a 'js' element which would point to a custom javascript file within your module.  The code in this file would watch elements of that CSS class for when a user hit 'tab', and then automatically click the 'Add More' button for you.  JS added below for reference:

(function ($) {
  Drupal.behaviors.formAutoTab = {
    attach: function (context, settings) {
      $('.othercustomclass').focus();
      $('.customclass').keydown(function (event) {
        if (event.keyCode == '9') {
          $('input[type=submit][id^=addbuttonidgoeshere]').mousedown();
          event.preventDefault();
        }
      });
    }
  }

})(jQuery);


The trick is .othercustomclass which you will need to refocus on the new element after it's added.  I think core Drupal does apply some sort of 'ajax-processed' class or something else to the fields added by the 'Add More' button, which you might be able to use as the jquery selector.