if I understand, you need to display category or taxonomy last child.
I think you should use get_terms function for this matter:
Get all first childrens from your taxonomy:
$my_childrens_firsts = get_terms( array('taxonomy' => 'your_taxonomy',
'hide_empty' => false,
'hierarchical' => true,
'parent'=> $your_cat_id,
) );
Now you have to foreach $my_childrens_firsts (foreach($my_childrens_firsts as $key => $first_children)… so you get your $first_children->term_id
still within this foreach and get next childs level, using again get_terms:
$my_childrens_seconds = get_terms( array('taxonomy' => 'your_taxonomy',
'hide_empty' => false,
'hierarchical' => true,
'parent'=> $first_children->term_id,
) );
and so on…until the sublevel you need.