Accessing Drupal.settings outside of attach function
Worked it out myself.

(function($) {
  Drupal.behaviors.myModule_someJS = {

    anotherFunction: function() {
      console.log(this.vars.initial_page);
    },

    attach: function(context, settings) {

      this.vars = {
        initial_page  : Drupal.settings.myModule.initial_page
      }

      this.anotherFunction();

    }
  }
})(jQuery);


Drupal.settings won't contain the custom stuff sent through from my module until the attach function is called, so therefore every settings variable can only be called inside attach()

(and from Alice's comments above, it may be a matter of drupal_add_js being called from within the page rather than in the init() hook)