Dielectric nanocavities are emerging as a versatile and powerful tool for the linear and nonlinear manipulation of light at the nanoscale. In this work, we exploit the effective coupling of electric and toroidal modes in AlGaAs nanodimers to locally enhance both electric and magnetic fields while minimizing the optical scattering, thereby optimizing their second-harmonic generation efficiency with respect to the case of a single isolated nanodisk. We also demonstrate that proper near-field coupling can provide further degrees of freedom to control the polarization state and the radiation diagram of the second-harmonic field.