Double-perovskite oxides have attracted recent attention due to their attractive functionalities and application potential. In this paper, we demonstrate the effect of dual controls, i.e., the deposition pressure of oxygen (P$_O2$) and lattice mismatch ($ε$), on tuning magnetic properties in epitaxial double-perovskite Sr$_2$FeReO$_6$ films. In a nearly-lattice-matched Sr$_2$FeReO$_6$/SrTiO$_3$ film, the ferrimagnetic-to-paramagnetic phase transition occurs when P$_O2$ is reduced to 30 mTorr, probably due to the formation of Re$^{4+}$ ions that replace the stoichiometric Re$^{5+}$ to cause disorders of $B$-site ions. On the other hand, a large compressive strain or tensile strain shifts this critical P$_O2$ to below 1 mTorr or above 40 mTorr, respectively. The observations could be attributed to the modulation of $B$-site ordering by epitaxial strain through affecting elemental valence. Our results provide a feasible way to expand the functional tunability of magnetic double-perovskite oxides that hold great promise for spintronic devices.