  We study the effect of combining spin fluctuations and forward scattering
electron-phonon ({\eph}) coupling on the superconductivity in the
FeSe/SrTiO$_3$ system modeled by a phenomenological two-band Hubbard model with
long-range {\eph} interactions. We treat the electron and phonon degrees of
freedom on an equal footing using a \emph{fully} self-consistent FLEX plus
Migdal-Eliashberg calculation, which includes a self-consistent determination
of the spin fluctuation spectrum. Based on FeSe monolayers, we focus on the
case where one of the bands lies below the Fermi level (i.e. incipient), and
demonstrate that the combined interactions can enhance or suppress $T_c$,
depending on their relative strength. For a suitable choice of parameters, the
spin-fluctuation mechanism yields a $T_c \approx 46.8$ K incipient $s_\pm$
superconductor, consistent with surface-doped FeSe thin films. A
forward-focused {\eph} interaction further enhances the $T_c$, as observed in
monolayer FeSe on SrTiO$_3$.
