  We consider the out-of-thermal-equilibrium Casimir-Polder interaction between
atoms of He$^*$, Na, Cs, and Rb and a cavity wall made of sapphire coated with
a vanadium dioxide film which undergoes the dielectric-to-metal phase
transition with increasing wall temperature. Numerical computations of the
Casimir-Polder force and its gradient as the functions of atom-wall separation
and wall temperature are made when the latter exceeds the temperature of the
environment. The obtained results are compared with those in experiment on
measuring the gradient of the Casimir-Polder force between $^{87}$Rb atoms and
a silica glass wall out of thermal equilibrium. It is shown that the use of
phase-change wall material increases significantly the force magnitude and
especially the force gradient, as opposed to the case of dielectric wall.
