While the youngest known supernova remnants, such as Cassiopeia A, have been proven to be able to accelerate cosmic rays only up to $\sim$10$^{14}\,\mathrm{eV}$ at their present evolutionary stages, recent studies have shown that particle energies larger than a few PeV ($10^{15}\,\mathrm{eV}$) could be reached during the early stages of a core-collapse Supernova, when the high-velocity forward shock expands into the dense circumstellar medium shaped by the stellar progenitor wind. Such environments, in particular the type IIn SNe whose progenitors may exhibit mass-loss rates as high as $10^{-2}M_\odot\,\mathrm{yr}^{-1}$\cite{smith14}, could thus lead to $\gamma$-ray emission from $\pi^0$ decay in hadronic interactions, potentially detectable with current Cherenkov telescopes at very-high energies. Such a detection would provide direct evidence for efficient acceleration of CR protons/nuclei in supernovae, and hence new insights on the long-standing issue of the origin of Galactic Cosmic Rays. In that context, the High Energy Stereoscopic System (\hess) has been carrying out a Target of Opportunity program since 2016 to search for such an early very-high-energy $\gamma$-ray emission towards nearby core-collapse supernovae and supernova candidates (up to $\sim 10~\mathrm{Mpc}$), within a few weeks after discovery. After giving an overview of this \hess Target of Opportunity program, we present the results obtained from the July 2019 observations towards the transient \at, originally classified as a type IIn supernova, which occurred in the galaxy M74 at $\sim 9.8\,\mathrm{Mpc}$. Although its nature still remains unclear, the derived \hess constraints on this transient are placed in the general context of the expected VHE $\gamma$-ray emission from core-collapse supernovae.