The oxygen stoichiometry variation in the BaRE(Cu0.5Fe0.5)2O5+δ series is established with rare earth elements having ionic radii (CN = 8) in the range of 0.977-1.160 Å (Lu-La). The double-perovskite structure is stable with RE = Lu-Pr, while La prefers the simple-perovskite structure. In the double-perovskite structures with RE = Lu-Sm the oxygen content was found to be constant (δ ≈ 0) against normal-pressure oxidative and reductive heat treatments, whereas a larger rare earth ion, i. e. Nd or Pr, allows continuous oxygen content tuning within about 0.23 and 0.32 oxygen atoms, respectively. In the La perovskite the oxygen content could be varied from δ ≈ 0 to 0.45. The 57Fe Mössbauer spectroscopy measurements revealed that the transition temperature of magnetic ordering decreased with an increase in the oxygen content for both the double- and simple-perovskite structures.