We experimentally investigate supercurrent through Coulomb islands, where island and leads are fabricated from semiconducting nanowires with fully surrounding superconducting shells. Applying flux along the wire yields a series of destructive Little-Parks lobes with reentrant supercurrent. We find Coulomb blockade with $2e$ peak spacing in the zeroth lobe and $1e$ average spacing, with regions of significant even-odd modulation, in the first lobe. Evolution of Coulomb-peak amplitude through the first lobe is consistent with a theoretical model of supercurrent carried predominantly by zero-energy states in the leads and the island.

• Bound states
• Coulomb blockade
• Density of states
• Josephson effect
• Superconductivity
• Josephson junctions
• Nanowires
• Quantum dots