  The quest to govern the driving forces behind superconductivity and gain
control over the superconducting transition temperature $T_{c}$ is as old as
the phenomenon itself. Microscopically, this requires a proper understanding of
the evolution of electron-lattice interactions in their parameter space. We
report such a controlled study on $T_{c}$ in In$_{x}$Te via fine-tuning the In
stoichiometry $x$. We find that increasing $x$ from 0.84 to 1 results in an
enhancement of $T_{c}$ from 1.3 K to 3.5 K accompanied by an increase of the
electron-phonon coupling constant from 0.45 to 0.63. Employing first-principles
calculations, we show that this behavior is driven by two factors, each taking
the dominant role depending on $x$. For $x\lesssim 0.92$, the major role is
played by the density of electronic states at the Fermi level. Above $x\sim
0.92$, the change in the density of states flattens while the enhancement of
$T_{c}$ continues. We attribute this to a systematic softening of lattice
vibrations, amplifying the electron-phonon coupling, and hence, $T_{c}$.
