I found that getting hair out of the drain works rather easy this way:
Take a thin wooden stick (I use the ones that are meant to skewer meat or veggies and then grill them), hold it into the drain and spin it around. The hair will cling to the wooden surface and wind itself around the stick.
So you'll reach easily hair that's already a bit down the drain and you won't even have to remove the sink strainer :o)

The more hair is in there, the better this works actually...