A detailed computational study of the wavelength-dependent efficiency of optical second-harmonic generation in plasmonic nanostructures is presented. The computations are based on a discontinuous Galerkin Maxwell solver that utilizes a hydrodynamic material model to calculate the free-electron dynamics in metals without any further approximations. Besides wave-mixing effects, the material model thus contains the full nonlocal characteristics of the electromagnetic response, as well as intensity-dependent phenomena such as the Kerr effect. To be specific, two prototypical nanostructures are studied in depth with the help of two independent computer codes. For an infinitely long metal cylinder, it is found that the spectral position of linear particle plasmon modes (dipolar modes, higher-order modes, and, for frequencies above the plasma frequency also bulk plasmon modes) and their associated relative strengths for scattering and absorption both at the fundamental and second-harmonic wavelengths largely control the conversion efficiency. Notably, Fabry–Perot resonances associated with longitudinal bulk plasmons may be detectable via background-free second-harmonic spectroscopy. For a more complex V-groove nanostructure, it becomes possible to engineer a doubly resonant scenario at the fundamental and the second-harmonic wavelength. This leads to an efficient enhancement of second-harmonic emission. Our work thus demonstrates that the careful design of nanostructures on the nonlocal linear level facilitates highly efficient nanoantennas for second-harmonic emission with applications in background-free imaging and frequency conversion systems.