  Superconductor-semiconductor hybrid systems play a crucial role in realizing
nanoscale quantum devices, including hybrid qubits, Majorana bound states, and
Kitaev chains. For such hybrid devices, subgap states play a prominent role in
their operation. In this work, we study such subgap states via Coulomb and
tunneling spectroscopy through a superconducting island defined in a
semiconductor nanowire fully coated by a superconductor. We systematically
explore regimes ranging from an almost decoupled island to the open
configuration. In the weak coupling regime, the experimental observations are
very similar in the absence of a magnetic field and when one flux quantum is
piercing the superconducting shell. Conversely, in the strong coupling regime,
significant distinctions emerge between the two cases. We ascribe this
different behavior to the existence of subgap states at one flux quantum, which
become observable only for sufficiently strong coupling to the leads. We
support our interpretation using a simple model to describe transport through
the island. Our study highlights the importance of studying a broad range of
tunnel couplings for understanding the rich physics of hybrid devices.
