Superconductivity in systems which lack inversion symmetry is expected to give rise to a host of unconventional effects, including mixed parity ground-states, and the formation of highly topological surface states. Our recent focus has been on the detection of time-reversal symmetry breaking in transition metal alloys, as this is one of the clearest signals of unconventional superconductivity. In this proposal, we suggest an investigation of two members in the family of Rhenium based alloys that crystallize in the non-centrosymmetric alpha-Mn structure. This will build upon our previous work in this family, and will allow us to build up a picture of how the strength of the spin-orbit coupling affects the parity and pairing symmetry of the superconducting ground state.