Unless I'm missing something, I reckon there's a really obvious improvement on that, but Stack Exchange doesn't let new users leave comments. Think of the old puzzle where you've got 9 bags of gold coins which weight 10 grams each and one bag of shaved coins which weigh only 9 grams each and you need to figure out which bag has the shaved coins in only one weighing. You should be able do the same thing here - add one "Cookie: secret=A", two "Cookie: secret=B" and so on. Because of the limited look-back window in the compressor you'll need to interleave them like ABCD...BCD...CD... but it should give more information.
Edit: Wait, I am. That wouldn't work because it'd compress the repeated subsequences. D'oh.
Probably, yeah. Thinking about it, one obvious approach might be to try and overlap the last step of guessing one byte with the first step of guessing the next byte, for instance.
Edit: Wait, I am. That wouldn't work because it'd compress the repeated subsequences. D'oh.