Resonant plasmonic metasurfaces have attracted much attention for great potential in augmenting nonlinear optical conversion at the nanoscale and thus related sensing and integrated optics applications. In this work, we use the nonlinear scattering theory to numerically investigate enhanced second-harmonic generation (SHG) from Fano metasurfaces which consist of gold asymmetric double-bars. We find that the Fano resonance at the fundamental wavelength boosts the nonlinear response by more than a factor of 60. On this basis, by introducing translational and rotational structural asymmetries, the SHG signal is further amplified because of the broken mirror symmetry. More specifically, under the optimal condition, the previously suppressed SHG component can be greatly released and play a more important role compared to the original existing SHG component in an extra 6-fold enhancement in total SHG intensity. The 360-fold enhancement by tailoring both resonance quality and structural asymmetries indicates the clear and important roles of both linear resonance and local-field distribution in reaching the largest SHG emission. Our results are a step towards enlarging SHG responses of more complex plasmonic nanostructures.