We have recently proposed the idea that dark matter in our universe is formed by primordial black holes generated by Standard Model Higgs fluctuations during inflation and thanks to the fact that the Standard Model Higgs potential develops an instability at a scale of the order of 1011 GeV. In this sense, dark matter does not need any physics beyond the Standard Model, although the mechanism needs fine-tuning to avoid the overshooting of the Higgs into the dangerous AdS vacuum. We show how such fine-tuning can be naturally avoided by coupling the Higgs to a very heavy scalar with mass ≫1011 GeV that stabilises the potential in the deep ultraviolet, but preserving the basic feature of the mechanism which is built within the Standard Model.