Resonantly excited plasmonic gold nanoantennas are strong sources of third-harmonic (TH) radiation. It has been shown that the response originates from large microscopic nonlinearity of the gold itself, which is enhanced by the near-field of the plasmonic nanoantenna. To further enhance this response, one can incorporate nonlinear media into the near-fields of the nanoantenna, as an additional TH source. To obtain a significant contribution from the added medium, its nonlinear susceptibility should be comparable to that of the antenna material. Many organic materials offer the necessary nonlinear susceptibility and their incorporation is possible with simple spin-coating. Furthermore, organic materials are often susceptible to photodegradation. This degradation can be used to investigate the influence of organic materials on the hybrid system. Our investigated hybrid organic plasmonic nanoantenna system consists of a gold nanorod array and poly(methyl methacrylate) as the nonlinear dielectric medium. The experiments clearly reveal two contributions to the generated TH radiation, one from the nanoantenna itself and one from the polymer. The nonlinear response of the hybrid material exceeds the response of both individual constituents and opens the path to more efficient nanoscale nonlinear light generation.