The new 'information' is the number of times you're being woken up. Even though this information is not being told to you, you can implicitly use it by using the same strategy every time.
You should see it from the perspective of every awakening.
Imagine this. The experiment is repeated 10 times. On average the coin will have landed 5 times on heads, and 5 times on tails. This means that you'll be woken up 5 times while the coin landed on heads. However, you'll be woken up 500000005 (!) times while the coin landed on tails.
Knowing this, when you're woken up only in 5/500000010 of the scenarios the coin landed on heads.
You should see it from the perspective of every awakening.
Imagine this. The experiment is repeated 10 times. On average the coin will have landed 5 times on heads, and 5 times on tails. This means that you'll be woken up 5 times while the coin landed on heads. However, you'll be woken up 500000005 (!) times while the coin landed on tails.
Knowing this, when you're woken up only in 5/500000010 of the scenarios the coin landed on heads.