If you want to exclude older posts from your search results, you can using pre_get_posts.
The following code uses the date_query from WP_Query.
Only returns results after the 1/1/2013 using the WordPress search function.
add_filter( 'pre_get_posts', 'search_filter' );
function search_filter($query) {
if ( ! is_admin() && $query->is_main_query() ) {
if ($query->is_search) {
$query->set( 'date_query',
[ [
'after' => 'January 1st, 2013',
'inclusive' => true,
] ]
);
}
}
}
The code works with the WordPress search function and search widget included in all WordPress installations.