  Molybdenum ditelluride (MoTe$_2$) is attracting considerable interest since
it is the archetypal type-II Weyl semimetal and a candidate for topological
superconductivity. We investigate the superconducting phase diagram of two
MoTe$_2$ polymorphs using the $ab~initio$ anisotropic Migdal-Eliashberg theory,
and we show that the superconducting dome originates from the synergistic
contribution of the density of states at the Fermi level and the transverse
acoustic Te modes in the 1T$^\prime$ phase. We find that the electron and hole
pockets carry trivial $s$-wave order parameters of slightly different
magnitude, reminiscent of a two-gap structure as suggested by recent
experiments. We suggest that a possible route for enhancing the superconducting
critical temperature, and realizing $s_{+-}$ pairing, in the T$_d$ phase is to
exploit its non-trivial band topology via electron doping.
