Of course, if the pattern is not found (or found late), your solution still slurps the whole file. The following solution keeps at most {buffer size}+{pattern length)-1 bytes of the file in memory at a time.

my $pattern = "\xFF\xD9";

open(FILE, "< $file")
or die("Unable to open input file: $!\n");

binmode(FILE);   # Disable "\n" translation.

$/ = \4096;      # Arbitrary buffer size.

my $block;
my $partial = '';
my $base = 0;

my $matches = 0;

while ($block = <FILE>) {
   my $lookbehind = $partial . $block;

   my $pos = -1;

   for (;;) {
      $pos = index($lookbehind, $pattern, $offset+1);
      last if $pos == -1;

      print("Found a match at ", $base+$pos-length($partial), "\n");
      $matches++;
   }

   $partial = substr($block, 1 - length($pattern));
   $base += length($block);
}

if ($matches) {
   print("Found $matches matches.\n");
} else {
   print("Pattern not found.\n");
}
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Will report overlapping matches.

Untested.