  Realizing topological superconductivity by integrating
high-transition-temperature ($T_C$) superconductors with topological insulators
can open new paths for quantum computing applications. Here, we report a new
approach for increasing the superconducting transition temperature
($T_{C}^{onset}$) by interfacing the unconventional superconductor Fe(Te,Se)
with the topological insulator Bi-Te system in the low-Se doping regime, near
where superconductivity vanishes in the bulk. The critical finding is that the
$T_{C}^{onset}$ of Fe(Te,Se) increases from nominally non-superconducting to as
high as 12.5 K when $Bi_2Te_3$ is replaced with the topological phase
$Bi_4Te_3$. Interfacing Fe(Te,Se) with $Bi_4Te_3$ is also found to be critical
for stabilizing superconductivity in monolayer films where $T_{C}^{onset}$ can
be as high as 6 K. Measurements of the electronic and crystalline structure of
the $Bi_4Te_3$ layer reveal that a large electron transfer, epitaxial strain,
and novel chemical reduction processes are critical factors for the enhancement
of superconductivity. This novel route for enhancing $T_C$ in an important
epitaxial system provides new insight on the nature of interfacial
superconductivity and a platform to identify and utilize new electronic phases.
