You clearly want to slurp the whole file

What if the file is hundreds of megs? Here's a little something I whipped up:

use strict;
use warnings;

my $buf_size = 16_384;
open(my $big, shift) or die;
open(my $small, shift) or die;
my $search_string;
{
        local $/;
        $search_string = <$small>;
}
my $buffer = "";
my $pos = 0;
while(sysread($big, $buffer, $buf_size, length($buffer) ) ) {
   if ( (my $index = index($buffer, $search_string)) != -1)  {
      print "FOUND! found the search string at position ", $pos + $ind
+ex;
      exit;
   }
   $buffer = substr($buffer, int(length($buffer)/2));
   $pos += length($buffer);
}

print "search string not found";
[download]

I'll grant that this also falls prey to the same problem that your has if the smaller file is itself large, but I think that in general this scales much better.