[hatthew]

A general direction you could go is to use blocks greater than 2. For example, you flip the coin exactly 64 times, and discard the result unless there is exactly 1 tails and 63 heads. This happens about 22% of the time, so it's on average 290 flips to get a single sample. When you do get that sample, you convert the single tails' position within that 64 block into binary, and get a 6 bit number uniformly distributed 0-63, i.e. you get 6 bits of entropy. So on average 6/290=0.02 bits of entropy per flip, twice as good as using blocks of 2, though only a quarter as good as the theoretical upper bound.

I picked "block of 64 with only a single tails" since it was simple, and I'm sure a mathematician could figure out how to optimize it much more, but my general point is to agree that there's definitely ways to get closer to the theoretical upper bound you mentioned.