The code change won't make any difference to a "hang", but will hide errors that are reported by the calling code dieing. Generally hiding such errors is a bad thing so I wouldn't do that. The following trivial code works fine fetching your problematic page:

use strict;
use warnings;
use LWP::Simple;

my $page = get('http://www.aaaedm.com/contact.htm');

print $page;
[download]

Maybe there are warnings or errors earlier in your code execution that are causing the problem you see at the point in the code you describe. Can you whittle your current script down to a half dozen or so lines that shows the issue for the URL you give?