I guess you’re referring to the extension I wrote to add a simple search to the user page?
If yes, then you can change that script to use the html filter that the ranking shortcode has. It is this part:
add_filter( 'footballpool_ranking_page_html', array( __CLASS__, 'add_search' ) );
You’ll have to look up the exact value in the shortcode class file. The rest of the logic can stay the same as both page and shortcode use the same functions to get the ranking.
If you want to have a search on both, then don’t change the line, but add a same one, but then with the filter from the shortcode.