To get you started.

Ostensibly, you only need to add 4 lines to your existing code (here cleaned up a little) to make it a threaded socket server:

#!/usr/bin/perl

use IO::Socket;
use threads;                             ## 1
use strict;

my $sock = new IO::Socket::INET (
    LocalHost => '127.0.0.1',
    LocalPort => '6590',
    Proto => 'tcp',
    Listen => 1,
    Reuse => 1,
) or die "Could not create socket: $!\n";

while(1){
    my $new_sock = $sock->accept();
    print $sock->connected();
    async {                          ## 2
        while(<$new_sock>) {
            print $_;
        }
        close $new_sock;
    }->detach;                       ## 3
    sleep 0;                         ## 4
}
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Note: That is untested, and not a perfectly working socket server; but pretty much everything that needs adding; also needs adding to your original code.