Writing to a closed socket causes a SIGPIPE, which is a signal, not an exception. So you have to handle that somewhow, ignore is usually simplest. Then writing to a closed socket becomes just a normal I/O error. Assuming we don't have to worry about the speed at which I/O progresses, the code becomes:

#! /usr/bin/perl -w
use strict;
use IO::Socket::INET;

$SIG{PIPE} = "IGNORE";
my $client;

sub call_server {
    my $answer;
    for (0..1) {
        return $answer if 
            $client and 
            print $client "some_command\n" and
            defined($answer = <$client>);
        $client = IO::Socket::INET->new(PeerAddr => "localhost",
                                        PeerPort => 16666) or
            die "Couldn't connect to server: $!";
    }
    die "Server in trouble";
}

foreach(1..10) {
   print call_server();
   sleep 10; # allow sometime to test server restart
}
[download]

(warning, only minimally tested)

updated: code didn't take into account that a write might still get out, and only the read will notice the close. Also be more paranoid about a bad server, only retry once and assume an immediate second failure is a sign of server problems.

updated: make the looping on first call consistent too.