Looks like it's known problem in perl.
Also, similar issue Can't catch signals after an exec?

Code can be simplified to

$sub = sub { exec "perl $0" };
$SIG{INT} = $sub;
$SIG{USR1} = $sub;

while(1) {
    print ++$counter, "\n";
    sleep 1;
}
[download]

(if you press Ctrl-C, it restarts only once)
Workaround for example in perldoc:

#!/usr/bin/env perl

    use POSIX ();
    use FindBin ();
    use File::Basename ();
    use File::Spec::Functions;
    $| = 1;
    # make the daemon cross-platform, so exec always calls the script
    # itself with the right path, no matter how the script was invoked
+.
    print STDERR "STARTED\n";
    my $script = File::Basename::basename($0);
    my $SELF = catfile($FindBin::Bin, $script);
    # POSIX unmasks the sigprocmask properly
    my $need_restart = 0;
    $SIG{HUP} = sub {
    print "got SIGHUP\n";
    $need_restart = 1;
    };
    code();
    sub code {
    print "PID: $$\n";
    print "ARGV: @ARGV\n";
    my $count = 0;
    while (++$count) {
    sleep 2;
    if ($need_restart) {
    exec($SELF, @ARGV) || die "$0: couldn't restart: $!";
    }
    print "$count\n";
    }
    }
[download]

(i.e. need to call exec() outside of signal handler)

And another workaround is posted above by McA Re: Using SIGHUP to restart a daemon