  We investigate the Little-Parks oscillations caused by a persistent current
loop set on the top edge of a mesoscopic superconducting thin-walled cylinder
with a finite height. For a short cylinder the Little-Parks oscillations are
approximately the same ones as the standard effect, as there is only one
magnetic flux piercing the cylinder. For a tall cylinder the inhomogeneity of
the magnetic field makes different magnetic fluxes pierce the cylinder at
distinct heights and we show here that this produces two distinct Little-Parks
oscillatory regimes according to the persistent current loop. We show that
these two regimes, and also the transition between them, are observable in
current measurements done in the superconducting cylinder. The two regimes stem
from different behavior along the height, as seen in the order parameter,
numerically obtained from the Ginzburg-Landau theory through the finite element
method
