Carefully use a test light on your cig lighter, or a multimeter and check for voltage. ground onto the ring, and poke the centre at the bottom, try not to bridge them or you will need to buy another fuse. If the voltage is there, then get a new lighter plug. If not, then check wiring. if it is all rusty and nasty insde, then you might need to buy a cig lighter socket replacement.
You can also try swtitching lighters between the back and front lighters, and see if the problem chases the plug, or the socket.
The cigarette lighter is "as new" since it's a non smoker's car. If the front lighter is out will that cause the rear one to go out also? Seems like they would be in tandem.