Use a Menu Component that is set to only display 1 item for your portlet. Then create a second content item that has a Presentation Template that just says something like this:
<div class="emptyPortlet" style="display:none;"> This is an empty template. </div>
*if you don't want to declare the style inside the tag, you could also specify .emptyPortlet { display: none;}
inside your CSS file.
Publish this before your other item and then set the Menu Component to display based on Publish Date. This way, when you create another item that you want to display (which uses a different Presentation Template), it will have a more recent publish date and will be shown on the page. But, once that is expired, the only published item left to find will be this blank one, and with "display: none" it will essentially hide the entire portlet from the user.