Sure. It's a bit tricky to wrap your head around it, because the HTTP::Response object you received contains the HTTP::Request object that was created when LWP encountered the (last) redirect. The request object contains the url you've been redirected to. It might be confusing to see that the request object can change, but it does so just as an added note, people shouldn't rely on it for example to determine the original url.

Anyway, you can access the url something like this:

To see the final uri:

my $response = $ua->...;
print($response->request()->uri(), "\n");
[download]

Or to see all the redirections:

my $response = $ua->...;
while ($response) {
   print($response->request()->uri(), "\n");
   $response = $response->previous();
}
[download]