I too have had that issue. When I use my power panel in the back (the jack access door) for the fridge, I will usually wedge something soft against it.
My opinion, cigarette sockets are a very poor way to get 12 VDC.
I've also used a a thin shim to make the plug fit tighter in the socket.
From some bubble pack or water bottle, cut a thin long piece of plastic, about a 1/4" wide x 1-1/2" long strip (like a large paper match). Place that next to the plug with a little longer on the bottom to lead in and then plug it in. It should now be nice and snug.
About the not connecting issue:
- Did you try rotating the plug in the socket?
- Is there any corrosion on the plug?
- Are the contact tabs on the plug sticking out far enough with a lot of spring force?
- Does it work in one position and not another? If so there may be a short in the plug.
Suggestion, if you do get another, perhaps get a combination USB and cigarette port. I just the dual USB port on the back of my console because I considered that the only persons to use it would be my daughter (when she was younger, and folks who need to charge their phones. I have since considered adding a cigarette socket there now since I don't really care for having to always feed the fridge line from the passenger middle seat to the driver's rear quarter panel.