Second-harmonic generation (SHG) in nanostructures gives rise to many applications such as lab-on-a-chip and imaging by frequency doubling. However, the SHG signal decreases with volume, and the conversion efficiency is limited. Thus, means to enhance nonlinear signals at the nanoscale are needed. For instance, while plasmonic nanostructures offer a high enhancement due to the strong confinement of the electromagnetic field, they have high losses and the fabrication methods are difficult. In this work, we propose to enhance the SHG by using the intrinsic scattering properties of an all-dielectric perovskite nanostructure. We demonstrate the Mie scattering resonances of individual barium titanate (BaTiO₃) nanoparticles with diameters between 200 and 250 nm. We distinguish contributions of the magnetic dipole and magnetic quadrupole. Then, we use the Mie resonances to achieve an SHG enhancement of 4 orders of magnitude within the same nanoparticle. Our results suggest that a strong increase of the SHG signal can be obtained without using plasmonic or hybrid nanostructures. We show a straightforward way of enhancing low optical signals within a single material, which will facilitate the study of other nonlinear phenomena at the nanoscale.