  The renormalization-scheme and scale dependence of the truncated QCD
perturbative expansions is one of the main sources of theoretical error of the
standard model predictions, especially at intermediate energies. Recently, a
class of renormalization schemes, parametrized by a single real number $C$, has
been defined and investigated in the frame of the standard perturbation
expansions in powers of the coupling. In the present paper we investigate the
$C$-scheme variation of a Borel-improved QCD perturbation series, which
implements information about the large-order divergent character of
perturbation theory by means of an optimal conformal mapping of the Borel
plane. In the new expansions, the powers of the strong coupling are replaced by
a set of expansion functions with properties which resemble those of the
expanded correlators, having in particular a singular behavior at the origin of
the complex coupling plane. On the other hand, the new expansions have a tamed
increase at high orders, as demonstrated by previous studies in the
$\overline{\text{MS}}$ renormalization scheme. Using as examples the Adler
function and the hadronic decay width of the $\tau$ lepton, we investigate the
properties of the Borel-improved expansions in the $C$-scheme, in comparison
with the standard expansions in the $C$-scheme and the expansions in
$\overline{\text{MS}}$. The variation with the renormalization scale and the
prescription for the choice of an optimal value of the parameter $C$ are
discussed. The good large-order behavior of the Borel-improved expansions is
proved also in the $C$-scheme, which is a further argument in favor of using
them in applications of perturbative QCD at intermediate energies.
