What helped is to move the functionality to modules, because IMHO modules are much neater to test than scripts.

Keep the script, but encapsulate all code in subroutines except the last line below. The last line ensures that one can require the script without actually running its' contents. I argue that this script is no harder to test than a module.

# script.pl
sub foo { ...; $retval; }
sub bar { ...; $retval; }

sub main {
  foo() or die();
  bar() or die();
}

# Call main only if no stack.  Non-empty stack means we're testing.
# See chapter 9 of Perl Testing - A Developer's Notebook
main() unless caller;
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