Non-centrosymmetric superconductors (NCSCs) are promising candidates in the search for unconventional and topological superconductivity. The α-Mn-type rhenium-based alloys represent excellent examples of NCSCs, where spontaneous magnetic fields, peculiar to time-reversal symmetry (TRS) breaking, have been shown to develop in the superconducting phase. By converse, TRS is preserved in many other isostructural NCSCs, thus leaving the key question about its origin fully open. Here, we consider the superconducting Re_1-xMo_x (0 ≤ x ≤ 1) family, which comprises both centro- and non-centrosymmetric structures and includes also two extra superconducting phases, β-CrFe and bcc-W. Muon-spin relaxation and rotation (μSR) measurements show a gradual increase of the relaxation rate below T_c, yet its independence of the crystal structure, suggesting that rhenium presence and its amount are among the key factors for the appearance and the extent of TRS breaking in the α-Mn-type NCSCs. The reported results propose Re_1-xMo_x as an ideal test case for investigating TRS breaking in unconventional superconductors.