  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 2$e$ peak spacing in the zeroth lobe and 1$e$ 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.
