Just to be clear first of all, Oyster Pay-as-you-go cannot be used for travel to or from Gatwick - I know that's not what the OP is asking about, but in case someone searches and finds this thread then I think it's worth stating that clearly.

To address the OP's question - yes, if you have a Travelcard (whether issued on Oyster or as a paper/printed ticket it makes no difference) you can buy extension tickets from the edge of the last zone it covers out to destinations outwith London - these are called 'Boundary Zone' tickets by the railway, and are issued from the outer boundary of a particular zone - so as a holder of a zones 1-6 Travelcard, you'd be issued with a ticket from Boundary Zone 6 to Gatwick. You have to go to the ticket office to buy one of these.