Any way to make custom taxonomy field searchable?

Here’s the code. You can change $post_type and $custom_fields according to your needs.

function extend_admin_search( $query ) {

    // Extend search for document post type
    $post_type="document";
    // Custom fields to search for
    $custom_fields = array(
        "_file_name",
    );

    if( ! is_admin() )
        return;

    if ( $query->query['post_type'] != $post_type )
        return;

    $search_term = $query->query_vars['s'];

    // Set to empty, otherwise it won't find anything
    $query->query_vars['s'] = '';

    if ( $search_term != '' ) {
        $meta_query = array( 'relation' => 'OR' );

        foreach( $custom_fields as $custom_field ) {
            array_push( $meta_query, array(
                'key' => $custom_field,
                'value' => $search_term,
                'compare' => 'LIKE'
            ));
        }

        $query->set( 'meta_query', $meta_query );
    };
}

add_action( 'pre_get_posts', 'extend_admin_search' );