Thanks for that hint. I now use IO::Interface to get my local ppp0 ip-address:

use IO::Socket;
use IO::Interface qw(:flags);

sub get_local_addr {
  my $if = shift;

  my $s = IO::Socket::INET->new(Proto => 'udp');
  unless ($s->if_flags($if)  & IFF_RUNNING) {
    return undef;
  } else {
    return $s->if_addr($if);
  }
}
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and then in my script I test, if the ip-address of the ppp0 interface changed:

use Net::Ping;
my $addr = get_local_addr("ppp0");
unless(defined $addr) {
  die "I'm offline\n";
}

while (1) {
  my $p = Net::Ping->new(udp);
  if $p->ping($host) {
    print "success\n";
  } else {
    print "failed\n";
    my $cur_addr = get_local_addr("ppp0");
    if ($addr ne $cur_addr) {
      die "I'm offline\n";
    }
  }
  undef $p;
  sleep 3;
}
[download]

There's no need to restart the script, because it will get started by my if-up script.