Less than 2 inches, in a lined cargo van. Not sure if the wagon is any deeper. You might want to look at surface mount sockets. I believe I've seen some, cruising Amazon, that are wedge-shaped... maybe.
Looks about right. And, if you did happen to find there's enough room, you could pull the socket out of the housing, and mount it through a hole.Found one!
BANDC Waterproof Marine Motorcycle Cigarette Lighter Power Socket 12v Power Outlet Tent https://www.amazon.com/dp/B00LZXFVXC/ref=cm_sw_r_cp_awd_pS6XwbQRH94CE