Use a sliding window:

my @lines;
$/ = \80; # assume a block size of 80

while (<$file>) {
  push @lines, $_;

  my $str = join "", @lines;
  if ($str =~ /searchword/) {
    my $loc = tell $file - pos $str;
    print "Found a match starting after $loc.\n";
  };

  if (@lines > 2) {
    shift @lines
  };
};
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Instead of looking for a match in just one "line", you look for a match in the "line" and the "line" after it.