  We study the Little-Parks effect for mesoscopic loops with very nonuniform
thickness. The results follow the trend of the phase diagram obtained for
almost uniform thickness. In particular, the singly-connected state is stable
on a line segment delimited by two critical points. Most of this study
considers loops with piecewise constant thickness; in this case the
Euler-Lagrange equation can be integrated analytically. Under appropriate
conditions, the temperature range where the singly-connected state is stable is
proportional to the square of the ratio between the maximal and the minimal
thicknesses. Our results may serve as a guide for planning experiments.
