Not a module, but...

#!/usr/bin/perl
use strict;
use warnings;
use 5.012;

BEGIN: recover_var();

# main here
# ... insert your existing code for getting the value here
# Since you didn't tell us the variable name, we'll use $var and assum
+e you've used it

my $var;

say "(\t\t Simulating execution of existing...";
say "\t\t code to obtain a (new) value for \$var";
say "\t\t and NOW simulating that said execution is complete. \t)\n";
$var += int(rand(113));

say "\n\tTake a pencil and write the current value of \$var, $var, on 
+your yellow sticky.";
say "\t Otherwise, the value of \$var will disappear when the RAM used
+ by \n\t this script is returned to the OS on termination of the scri
+pt./n";
say "\n\t\t\tTerminating...";
exit;


sub recover_var {
  if ( !$var ) {
    print "\n\t Find your yellow sticky (you did write down the old va
+lue, didn't you?) \n\t and enter the last \$var found on the sheet\n\
+t or '0' if no sticky is found: ";
    chomp ($var = <>);
    say "\n The last known value of \$var was $var.\n";
    return;
  }
}
[download]

Tested, but may not be suitable for purchaser's application. No warranty of fitness implied.

Despite the actual syntax error (the begin block that isn't), this passed perl -c ... with flying colors. Expanding on the note above, this is NOT suitable for OP's need, but taken litterally, OP asked for a way to keep a variable in memory after script termination... so here's a way, albeit with "dead tree RAM".

:-)