Although supernova remnants are believed to be the most plausible sources of galactic cosmic rays (CRs), it is yet unclear how they can accelerate particles beyond 1 PeV, especially when they evolve in the warm phase of the interstellar medium (ISM). On the other hand, extra-galactic sources are not expected to contribute below hundreds of PeV. This indicates a glaring gap in our understanding of the CR spectrum observed near the Earth. In this work, we propose a new model for the origin of Galactic CRs, taking into consideration the fact that most core-collapse supernovae explode within massive star clusters, and therefore their remnants do not expand in the warm ISM, but rather in a region where magnetic fields are amplified by the nearby powerful stellar wind outflows. By analyzing the population of young star clusters in the Milky Way, we find that a fraction of the supernova remnants expanding around the compact cluster cores are able to accelerate protons beyond 10 PeV. We show that particle acceleration around these clustered supernova remnant shocks can explain the observed CR spectrum up to several hundreds of PeV.