Cache breaks functionality?
I have redone my code as suggested in the comments so that it uses Drupal behaviors and the Drupal's javascript-api. I also moved my code out of the existing theme's script.js to its own file, and added that file to the scripts array in the theme's .info file.

Below is my updated code, in case it should help anyone that encounters this same sort of problem and is unfamiliar with Drupal behaviors. This has solved my problem with my code no longer working once the JS is aggregated and/or cached.

   (function($) {
     Drupal.behaviors.contactLinkHoverPopout = {
         attach: function(context, settings) {
             var div = '&lt;div class="contact-popup"&gt;' +
                 '10320 Globe Road&lt;br/&gt;Morrisville, NC 27560&lt;br/&gt;' +
                 'Phone: &lt;a href="tel:919-361-4333"&gt;919-361-4333&lt;/a&gt;&lt;br/&gt;' +
                 'Fax: 919-361-4388&lt;br/&gt;' +
                 '&lt;a href="mailto:info@dosoffice.com"&gt;info@dosoffice.com&lt;/a&gt;';

             var popupVisible = false;

             // we only want mouseovers applied once!   
             $('a.contact-link', context).once(function() {
                 // add the invisible popup div
                 $('body', context).append(div);

                 // add mouseover to contact-link
                 $('a.contact-link', context).mouseover(function() {
                     $('div.contact-popup', context).css('left', event.pageX + "px");
                     $('div.contact-popup', context).css('top', event.pageY + "px");
                     $('div.contact-popup', context).css('visibility', 'visible');
                     popupVisible = true;
                 });

                 $('div.contact-popup', context).mouseover(function() {
                     $('div.contact-popup', context).css('visibility', 'visible');
                     popupVisible = true;
                 });

                 $('a.contact-link', context).mouseout(function() {
                     if (popupVisible) {
                         $('div.contact-popup', context).css('visibility', 'hidden');
                         popupVisible = false;
                     }
                 });

                 $('div.contact-popup', context).mouseout(function() {
                     if (popupVisible) {
                         $('div.contact-popup', context).css('visibility', 'hidden');
                         popupVisible = false;
                     }
                 });
             });
         }
     };
 }(jQuery));