How can I efficiently dig small footing holes?
For that size hole, I'd probably go with a Post Hole Digger.  If it's sharp and there aren't too many rocks or roots, you just shove it into the ground, pull the handles apart, and lift.

However, for the number you're going to be digging, it might be better to visit your local tool rental shop, and rent a gas powered auger.  If you can get a second person, the two person models are a lot easier to control ... you can then square it off with a shovel, and either get the last bit out by hand, or with a post hole digger.