I think you changed the question. Before you go to sleep you know you have a 50% chance of waking up to heads and a 50% chance of waking up to tails twice. When you wake up you haven't learned anything so those probabilities can't have changed.
You have arguably less information when you wake up in that you don't know the time. Are you waking up because it's tails? because it's heads? or is it tails the second time?