Custom rewrite after the default term link URL

(Revised answer)

So Let’s See The Issues With Your Rewrite Rule..

1. The first parameter passed to add_rewrite_rule() (which is the regular expression pattern) should not start with home_url() or any functions which outputs the website URL. So this is wrong:

   home_url() . '/car/([^/]*)/([^/]*)/?'
   

   because the generated regular expression (RegEx) pattern would be:

   http://example.com/car/([^/]*)/([^/]*)/?
   

   which will never be matched, because the request (see WP::$request) never starts with the website URL. And it doesn’t even start with /.

2. Is the &model=$matches[1] part a typo?

   index.php?taxonomy=car&term=$matches[1]&model=$matches[1]&trim=$matches[2]
   

Now The Fixes

1. The home_url() . '/car/([^/]*)/([^/]*)/?' should be (no trailing / at the start):

   '^car/([^/]*)/([^/]*)/?'
   

2. The index.php?taxonomy=car&term=$matches[1]&model=$matches[1]&trim=$matches[2] should probably be without the &model=$matches[1]:

   index.php?taxonomy=car&term=$matches[1]&trim=$matches[2]
   

   However, even with that &model=$matches[1], the rule actually worked in my case.

3. And not sure if you’ve already added this somewhere in your code, but I’d add it so that I could get the value of the “trim” parameter in the URL:

   add_rewrite_tag( '%trim%', '([^/]+)' );
   

   You’d add that before/after the add_rewrite_rule() line, and to get the “trim” value, you could use get_query_var( 'trim' ).