How can I retain get_post_type() in between consecutive searches after a search returns no results?

You can use get_query_var() to access the post type that was used for the search, and then add that as a hidden field on the search page.

Untested:

printf( 
    '<input type="hidden" name="post_type" value="%s" />', 
    get_query_var( 'post_type', get_post_type() ) 
);