How do allow an exposed filter to override another filter in Views?
You can modify the view paramteres implementing hooks_views_pre_build. There you can examine received data (for example, checking $_GET) and modify view filters, deleting the "Posted Date: Relative 'today'"filter.

function mymodule_views_pre_build(&amp;$view) {
  if ($view-&gt;name == 'my-view') {
    if (must_delete_filter()) {
      unset($view-&gt;filter['posted_date_relative_filter_name']);      
    }
  }
}


So Views is executed without that filter. In example code must_delete_filter is the function that decides if filter must be deleted or no.