So, require your module to be one of the first in the list. This worked for me:

runme.pl

#!/usr/bin/perl

# In order to 'use' a perlmodule in a userspace library you need to ad
+d it to @INC.
# But if you do not want to use an absolute path inside the script, yo
+u need this workaround:
BEGIN{ 
        use Cwd 'abs_path';
        my $this = $0; # this script 
        $this =~ s/[^\/]+$//; # strip the filename, now we have a (rel
+ative) directory
        my $path = abs_path($this); # now we have an absolute director
+y 
        unshift @INC, $path; # which we add to @INC
        # chdir('/tmp'); # does seem to confuse abs_path! Uncomment to
+ see
}


use My::Module;
My::Module::print_file_name();
My::Module::print_script_name();
My::Module::print_caller_name();
[download]

-*Module.pl*- ./My/Module.pm

#!/usr/bin/perl

package My::Module;
use feature qw(say);

local $_SCRIPT = "";
local $_PATH = "";
BEGIN{ 
        use Cwd 'abs_path';
        $_PATH = $0; # this script 
        $_PATH =~ s/([^\/]+)$//; # strip the filename, now we have a (
+relative) directory
        $_SCRIPT = $1; # the stripped part is the filename
        $_PATH = abs_path($_PATH); # now we have an absolute directory
+ 
         print "My::Module 0=$0 _PATH=$_PATH _SCRIPT=$_SCRIPT \n";
}

sub print_file_name {
    say __FILE__;
}

sub print_script_name {
    say "print_script_name: $_PATH/$_SCRIPT";
}


sub print_caller_name {
    my @D = caller();
    $D[1]= abs_path($_PATH.'/'.$D[1]) unless $D[1]=~m{^/};
    say "print_caller_name: @D";
}


1;
[download]

If you bet on the fact that if chdir() was done, and that new directory does NOT contain the script that was run, You can add a kill or warning in the BEGIN block of the Module.pm, like:

warn "ALERT ALERT ALERT" unless(-f "$_PATH/$_SCRIPT");
[download]

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