Brian,

After seeing your reply I went ahead and updated to 10.2 and you are absolutely correct when saying the whole process has changed in the welcome.aspx file. After playing around for a bit on how to accomplish what you are asking I decided to go the JavaScript / jQuery route. Below is the code (copy and paste it right before the </body> element) that can be used to modify portions of the Welcome page once all of the elements are loaded (this is due to the page now being rendered using Ajax calls). Just remember to replace the HTML code that I have in the append function with your own.

<script type="application/javascript" nonce="<%= PageNonce %>">
function waitForElement(selector) {
  return new Promise(function(resolve, reject) {
    var element = document.querySelector(selector);

    if(element) {
      resolve(element);
      return;
    }

    var observer = new MutationObserver(function(mutations) {
      mutations.forEach(function(mutation) {
        var nodes = Array.from(mutation.addedNodes);
        for(var node of nodes) {
          if(node.matches && node.matches(selector)) {
            observer.disconnect();
            resolve(node);
            return;
          }
        };
      });
    });

    observer.observe(document.documentElement, { childList: true, subtree: true });
  });
}	

waitForElement("#SearchFormDiv").then(function(element) {
	$('#SearchFormDiv').append('<div style="min-height:0px;max-width:700px;margin:auto;margin-top:30px;">'+ Date() +'</div>');
});
</script>