My suggestion would be to assign a variable to the length of the file prior to the first append. Then, just check the filesize; when it grows you know you've got more text to read:

    my $len = (-s "file.txt") || 0;
    system ("perl eck.pl $var > file.txt &");

    # ... later ....
    my $newlen = (-s "file.txt") || 0;

    if ($newlen > $len) {
        my @newlines = read_new_lines("file.txt", $len);
        $len = $newlen;  # Adjust length to new end of file
    }
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The subroutine read_new_lines() would look something like this:

    sub read_new_lines {
        my ($fn, $offset) = @_;
        use IO::File;
        my $fh = new IO::File($fn);
        $fh or die "Failed to read '$fn' ($!)\n";
        seek($fh, $offset, 1) or die "Failed seek ($!)\n";
        chomp(my @lines = <$fh>);
        return (@lines);
    }
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I use this trick a lot for webpages which need to monitor continuously updating logfiles.