For something this important I'd like to play it out but since that takes a while, think we should hash out some details beforehand. How much exactly Jack will reveal and how long he'll be captured. The most likely course of events from my end will be, after getting the snot kicked out of him Jack will be stuck in a bunker the base maintains and interrogated for a few hours by some soldiers loyal to the general, followed by a more personal face to face with the big bald guy himself, involving a monologue about how this is all for the best & the future of humanity, blah blah I bet if he wasn't reincarnated he'd be tired of that by now.

Then he'd be left to stew for a bit, which would give him time to get on the network.

AND as for Jack's family, nothing yet. Presently he's General Miller's problem and Albero is the guy who fucks with families. But expect it to be used against him in the future nonetheless.