Shortcode just with a php code

It sound like you can accomplish this with 2 PHP functions.

1) Save the searched taxonomy to the user’s session. When they re-run the search function, update the taxonomy that was searched. This way if its saved to their session, you can use it elsewhere on the site just for that user.

2) Create a shortcode to display that saved taxonomy.

// ... search function that gets $tt. 
save_searched_taxonomy($tt->name);
//...

function save_searched_taxonomy($taxonomy){
    $_SESSION['searched_taxonomy'] == $taxonomy;
}


function get_searched_taxonomy_shortcode(){
    return $_SESSION['saved_taxonomy'];
}
add_shortcode('searched_taxonomy', 'get_searched_taxonomy_shortcode');

Then on other pages, you can call your shortcode and have it refer to whatever that user searched for.

[searched_taxonomy]