How can I maximize ginger extraction?
To increase flavor extraction (and this applies to ALL flavors, not just ginger):


Simmer for a prolonged period (more time to extract flavor).
Puree or finely mince the ginger, then strain it out with a fine chinois or cheesecloth. Smaller pieces allow water to more easily penetrate, and allows flavor compounds to be extracted more quickly
Stir frequently; this greatly reduces the time you need to cook the ginger, by ensuring flavor compounds reach an equilibrium throughout the mixture, rather than being concentrated at the surface of the ginger pieces.
Cook the same ginger multiple times with fresh water each time.  This will extract more flavor than doing one batch cooked 3x as long, and also more than cooking the same ginger with 3x as much water.  Each batch will be less flavorful, of course, but after 3-5 batches, you should have most of the flavor out.
Reduce the ginger water after extraction to concentrate the flavor.  This will be especially helpful when combined with multiple batches. 
Add alcohol to the water, as many flavor compounds are oils, and thus more soluble in alcohol than water.  This is excellent to combine with reduction, as the alcohol is easily evaporated.


All of these approaches are based both practice and the chemistry behind extractions, which is what you are doing here.