  We analyze the subgap excitations and phase diagram of a quantum dot (QD)
coupled to a semiconducting nanowire fully wrapped by a superconducting (S)
shell. We take into account how a Little-Parks (LP) pairing fluxoid (a winding
in the S phase around the shell) influences the proximity effect on the dot. We
find that under axially symmetric QD-S coupling, shell fluxoids cause the
induced pairing to vanish, producing instead a level renormalization that
pushes subgap levels closer to zero energy and flattens fermionic parity
crossings as the coupling strength increases. This fluxoid-induced
stabilization mechanism has analoges in symmetric S-QD-S Josephson junctions at
phase $\pi$, and can naturally lead to patterns of near-zero modes weakly
dispersing with parameters in all but the zero-th lobe of the LP spectrum.
