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Single crystals of the new Ga-rich phases ReGa_~5(Sn), ReGa_~5(Pb) and ReGa_~5(Bi) were successfully obtained from the flux method. The new tetragonal phases crystallize in the space group <i>P4/mnc</i> (No. 128) with vertex-sharing capped Re₂@Ga₁₄ oblong chains. Vacancies were discovered on the Ga4 and Ga5 sites, which can be understood as the direct inclusion of elemental Sn, Pb and Bi into the structure. Heat capacity measurements were performed on all three compounds resulting in a small anomaly which resembles the superconductivity transition temperature from the impurity ReGa₅ phase. The three compounds were not superconducting above 1.85 K. Subsequently, electronic structure calculations revealed a high density of states around the Fermi level, as well as non-bonding interactions that likely indicate the stability of these new phases.