Nonlinear processes are at the core of many optical technologies whose further development require optimized materials suitable for nanoscale integration. Here we demonstrate the emergence of a strong bulk second-order nonlinear response in a plasmonic nanorod composite comprised of centrosymmetric materials. We develop an effective-medium description of the underlying physics, compare its predictions to the experimental results, and analyze the limits of its applicability. We demonstrate strong tunable generation of the p-polarized second-harmonic light in response to either s-orp-polarized excitation. High second-harmonic enhancement is observed for fundamental frequencies in the epsilon-near-zero spectral range. The work demonstrates emergence of structurally tunable nonlinear optical response in plasmonic composites and presents a new nonlinear optical platform suitable for integrated nonlinear photonics.