カスタムタクソノミーのセレクトボックスによる絞り込み検索を実装する方法

<form method="get" id="search-form" action="<?php echo esc_url(home_url('/')); ?>">
  <input type="hidden" name="post_type" value="投稿タイプ名">
  <input type="hidden" class="field" name="s">
  <select name="term_select">
    <?php
      $taxonomy_name = 'タクソノミー名';
      $taxonomys = get_terms($taxonomy_name);
      if(!is_wp_error($taxonomys) && count($taxonomys)):
      foreach($taxonomys as $taxonomy):
        $tax_posts = get_posts(array('post_type' => get_post_type(), 'taxonomy' => $taxonomy_name, 'term' => $taxonomy->slug ) );
      if($tax_posts):
    ?>
      <option value="<?php echo $taxonomy->slug; ?>"><?php echo $taxonomy->name; ?></option>
    <?php
      endif;
      endforeach;
      endif;
    ?>
  </select>
  <input type="submit" value="検索">
</form>

<?php
//絞り込みの値を取得
$s = $_GET['s'];
$post_type = $_GET['post_type'];
$term_select = $_GET['term_select'];
$post_tag = $_GET['post_tag'];
 
//絞り込みの値をクエリ用に代入
if( !empty($term_select) ) {
  $term_selected = array('taxonomy'=>'タクソノミー名','terms'=>$term_select,'field'=>'slug','operator'=>'IN');
}
 
if( !empty($post_tag) ) { //タグの場合
  $post_tag_selected = array('taxonomy'=>'post_tag','terms'=>$post_tag,'field'=>'slug','operator'=>'IN');
}
 
//タクソノミー絞り込みの場合はクエリを指定
if( !empty($term_select) || !empty($post_tag) ) {
  query_posts( array(
    'paged' => $paged,
    'post_type' => $post_type,
    's' => $s,
    'tax_query' => array(
      'relation' => 'AND',
      $term_selected,
      $post_tag_selected
    )
  )
  );
}
?>

<?php if( !empty($term_select) || !empty($post_tag) ) { ?>
  <div><strong>"
    <?php
      echo get_term_by('slug',$term_select,"タクソノミー名")->name." ";
      echo get_term_by('slug',$post_tag,"post_tag")->name;
    ?>
  "</strong> の検索結果</div>
<?php } else { ?>
  <div><strong>"<?php the_search_query(); ?>"</strong>の検索結果</div>
<?php } ?>

<ul>
  <?php if( have_posts() ): while ( have_posts() ) : the_post(); ?>
    <li><a href="<?php the_permalink(); ?>" title="<?php the_title_attribute(); ?>"><?php the_title(); ?></a></li>
  <?php endwhile; else: endif; ?>
</ul>

<?php
  $big = 9999999999;
  $arg = array(
    'base' => str_replace( $big, '%#%', esc_url( get_pagenum_link( $big ) ) ),
    'current' => max( 1, get_query_var('paged') ),
    'total' => $wp_query->max_num_pages,
    'mid_size' => 5,
    'prev_text' => '≪',
    'next_text' => '≫'
  );
  echo paginate_links($arg);
?>