That did the trick. I was missing
$ua->requests_redirectable ([]);
Thank you!

I do read the docs. I just don't understand what I am seeing, which I why I appreciate your help. (You will notice that I include my references in my code's comments)

#!/usr/bin/perl

use strict;
use warnings;
#use diagnostics;
#use diagnostics "-verbose";

use constant MaxTime1 => 20;   # Seconds
use LWP::UserAgent;
use Term::ANSIColor qw ( BOLD BLUE RED GREEN RESET );


my $Debug = "yes";
my $RedirectUrl;
my $IAm = "UrlRedirectTest.pl ";
my $WebSite;


sub GetUrlRedirect ( $$ ) {
   # Return $Url's redirected web page.  Black string if none if locat
+ed

   # References: 
   #   http://stackoverflow.com/questions/26087546/perl-print-the-redi
+rected-url
   #   http://search.cpan.org/~ether/libwww-perl-6.15/lib/LWP/UserAgen
+t.pm
   #   http://stackoverflow.com/questions/2470053/how-can-i-get-the-ul
+timate-url-without-fetching-the-pages-using-perl-and-lwp
   #   https://metacpan.org/pod/HTTP%3A%3AResponse
   #       $r->redirects
   #       Returns the list of redirect responses that lead up to this
+ response
   #       by following the $r->previous chain. The list order is olde
+st first.
   #       In scalar context return the number of redirect responses l
+eading up
   #       to this one.
   #
   # system ('curl -s "SomeUrl" -w %{redirect_url}');

   my $IAm     = ( caller(0) )[3];

   # Incoming:
   my $Url    = $_[0];
   my $Caller = "$_[1]" . "$IAm";

   # supposting variables
   my $ua;
   my $res;
#   my $redirect;
#   my $NewUrl;
   
   $ua = LWP::UserAgent->new;
   $ua->timeout ( MaxTime1 );
   $ua->show_progress;
   $ua->requests_redirectable ([]);

   $res = $ua->get($Url);

   # The parentheses around $redirect tell Perl that you want $redirec
+t
   # to be seen as list as you in fact are only interested in the firs
+t redirec

   if ($res->is_redirect) {
      my ( $NewUrl ) = $res->header('Location');
      # my $NewUrl = ( $res->header('Location') ) [1];
      if ( "$Debug" eq "yes" ) {
         print STDERR "Redirect Url = <${NewUrl}>\n"; }
         return ${NewUrl};
      }
   else { 
      print STDERR "${Caller}: ${Url} has no redirect\n";
      return "";
   } 
}


$WebSite = 'http://gbis.com';
$RedirectUrl = GetUrlRedirect ( $WebSite, $IAm );
print "WebSite <$WebSite> redirects to <$RedirectUrl>\n\n";

$WebSite = 'http://google.com';
$RedirectUrl = GetUrlRedirect ( $WebSite, $IAm );
print "WebSite <$WebSite> redirects to <$RedirectUrl>\n\n";

$WebSite = 'http://usa.kaspersky.com/files/?file=kis&program=one&lang=
+en&track=pu_kiskis_usen';
$RedirectUrl = GetUrlRedirect ( $WebSite, $IAm );
print "WebSite <$WebSite> redirects to <$RedirectUrl>\n\n";

END


$ ./Url.Redirest.Test.pl
UrlRedirectTest.pl main::GetUrlRedirect: http://gbis.com has no redire
+ct
WebSite <http://gbis.com> redirects to <>

Redirect Url = <http://www.google.com/>
WebSite <http://google.com> redirects to <http://www.google.com/>

Redirect Url = <https://klamericas.s3.amazonaws.com/files/one/en/kis17
+.0.0.611en_10755.exe>
WebSite <http://usa.kaspersky.com/files/?file=kis&program=one&lang=en&
+track=pu_kiskis_usen> redirects to <https://klamericas.s3.amazonaws.c
+om/files/one/en/kis17.0.0.611en_10755.exe>
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