You can use tell to get the line beginnings and store them in an array. Once you reach the end of the input, use seek to retrieve the lines:

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

my @positions;
open my $OUT, '+>', '2.out' or die $!;


for my $i (1 .. 20) {
    unshift @positions, tell $OUT;
    print {$OUT} "line $i\n";
}

open my $BACK, '>', '2.back' or die $!;
for my $pos (@positions) {
    seek $OUT, $pos, 0;
    my $line = <$OUT>;
    print {$BACK} $line;
}
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Update: Used the +> mode to keep the filehandle open.