I believe you can simply replace:

            do {
                next LINE;
                print RESULTS "$currentfile";
            } until eof();
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With print RESULTS <FILE>; because <FILE> will be treated in array context there and grab everything. After that, execution will fall through naturally and when it tries while ($currentfile=<FILE>) { again, it will be at eof so will break out of that loop.

Another approach (basically set a flag for when outputting is ok:

my $matchFound = 0;
while ($currentfile=<FILE>) {
  if( ! $matchFound ){
    $matchFound = 1 if $currentfile =~ /^\d{6}/ && $search_date eq sub
+str($currentfile,52,10);
  }
  next unless $matchFound;
  print RESULTS $currentfile;
}
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And yet another way (take advantage of split), though this involves slurping the log which might be bad if it's large (very untested):

my $currentfile = do { local $/ = undef; <FILE> };
my @parts = split /(^\d{6}).{46}$search_date)/m, $s, 2;
print RESULTS @parts[1,2] if $parts[1];
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