You possibly have overheated your lighter outlet and thus melted the
cheap plastic housing and losing either connecting ability or
contacts won't engage properly with your plug in to the lighter outlet.
Some of these are almost impossible to replace anymore with the
cheap or ignorant way they are assembled. Example: '99 Dodge
Durango lighter sockets will melt into their surrounding plastic bezel,
thus requiring replacement of the bezel and lighter outlets both. Not
to mention this being an obvious fire hazard. I would not run any
equipment from one of these sockets without first knowing the
amperage draw required and the maximum amperage allowed by
the manufacturer of the vehicle for the socket in question.