When publishing multiple lists then these can be selected via either a drop-down menu or box selector.

Drop Down

A dropdown is added above the lists, before any other drop-down filters within the list, to choose which list should be opened. If also filtering by Contest in the publish list settings then a dropdown for the Contests will be shown first. 

Box Selector

When opening the page a series of Boxes is shown for each Contest (if filtering by Contest) and a box for All Contests, with the available lists shown below, clicking a list will open the corresponding list.