  It has been proposed that the superconducting transition temperature
$T_{\mathrm{c}}$ of an unconventional superconductor with a large pairing scale
but strong phase fluctuations can be enhanced by coupling it to a metal.
However, the general efficacy of this approach across different parameter
regimes remains an open question. Using the dynamical cluster approximation, we
study this question in a system composed of an attractive Hubbard layer in the
intermediate coupling regime, where the magnitude of the attractive Coulomb
interaction $|U|$ is slightly larger than the bandwidth $W$, hybridized with a
noninteracting metallic layer. We find that while the superconducting
transition becomes more mean-field-like with increasing interlayer hopping, the
superconducting transition temperature $T_{\mathrm{c}}$ exhibits a nonmonotonic
dependence on the strength of the hybridization $t_{\perp}$. This behavior
arises from a reduction of the effective pairing interaction in the correlated
layer that out-competes the growth in the intrinsic pair-field susceptibility
induced by the coupling to the metallic layer. We find that the largest
$T_{\mathrm{c}}$ inferred here for the composite system is below the maximum
value currently estimated for the isolated negative-$U$ Hubbard model.
