In the general case, you can backtrack the response chain using the previous method for HTTP::Response objects. I got a lot of help in this example from the GET script that comes with LWP:

use LWP::UserAgent;

my $ua       = LWP::UserAgent->new;
my $url      = "http://imdb.com/find?nm=on;mx=20;q=eliza%20dushku";
my $response = $ua->get($url);
my @chain    = ( $response );

while ( $chain[0]->previous ) {
    unshift @chain, $chain[0]->previous;
}

for (@chain) {
    printf "%s %s --> %s\n" =>
        $_->request->method,
        $_->request->url->as_string,
        $_->status_line;
}

__END__

GET http://imdb.com/find?nm=on;mx=20;q=eliza%20dushku --> 302 Found
GET http://imdb.com/name/nm0244630/ --> 200 OK
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For your purposes, it's probably simpler to just compare the URL of the final HTTP::Response object with the one you gave to the UserAgent. If they're different, you must have encountered some sort of redirect:

my $response = $ua->get($url);

if ( $response->request->url->as_string ne $url ) {
    ...
}
[download]