Audio jack detection to turn on device without power loss
Not really for a stereo headphone, but you can do it for a mono jack by hooking the battery negative lead to the ring contact on a stereo socket... Classic guitar pedal trick. 

About the best you can do is a mosfet with a 22M resistor to bias the gate and use one of the headphone switches to pull the gate down when nothing is inserted. 

Regards, Dan.