An observation of neutron antineutron oscillations ( 𝑛−𝑛̅), which violate both 𝐵 and 𝐵−𝐿 conservation, would constitute a scientific discovery of fundamental importance to physics and cosmology. A stringent upper bound on its transition rate would make an important contribution to our understanding of the baryon asymmetry of the Universe by eliminating the post sphaleron baryogenesis scenario in the light quark sector. We show that one can design an experiment using slow neutrons that in principle can reach the required sensitivity of 𝜏𝑛−𝑛̅~1010 𝑠 in the oscillation time, an improvement of ~104 in the oscillation probability relative to the existing limit for free neutrons. The improved statistical accuracy needed to reach this sensitivity can be achieved by allowing both the neutron and antineutron components of the developing superposition state to coherently reflect from mirrors. For sufficiently small transverse momenta of 𝑛 𝑛̅ and for certain choices of nuclei for the 𝑛 𝑛̅ guide material, the relative phase shift of the 𝑛 and 𝑛̅ components upon reflection and the 𝑛̅ annihilation rate can be small enough to maintain sufficient coherence to benefit from the greater phase space acceptance the mirror provides. We point out a possible step by step implementation of the new method.