  Systems with an O(n) symmetrical Hamiltonian are considered in a
$d$-dimensional slab geometry of macroscopic lateral extension and finite
thickness $L$ that undergo a continuous bulk phase transition in the limit
$L\to\infty$. The effective forces induced by thermal fluctuations at and above
the bulk critical temperature $T_{c,\infty}$ (thermodynamic Casimir effect) are
investigated below the upper critical dimension $d^*=4$ by means of
field-theoretic renormalization group methods for the case of periodic and
special-special boundary conditions, where the latter correspond to the
critical enhancement of the surface interactions on both boundary planes. As
shown previously [\textit{Europhys. Lett.} \textbf{75}, 241 (2006)], the zero
modes that are present in Landau theory at $T_{c,\infty}$ make conventional
RG-improved perturbation theory in $4-\epsilon$ dimensions ill-defined. The
revised expansion introduced there is utilized to compute the scaling functions
of the excess free energy and the Casimir force for temperatures
$T\geqT_{c,\infty}$ as functions of $\mathsf{L}\equiv L/\xi_\infty$, where
$\xi_\infty$ is the bulk correlation length. Scaling functions of the
$L$-dependent residual free energy per area are obtained whose $\mathsf{L}\to0$
limits are in conformity with previous results for the Casimir amplitudes
$\Delta_C$ to $O(\epsilon^{3/2})$ and display a more reasonable
small-$\mathsf{L}$ behavior inasmuch as they approach the critical value
$\Delta_C$ monotonically as $\mathsf{L}\to 0$.
