Rhenium oxides belonging to the family AReO4 where A is a metal cation, exhibit interesting electronic and magnetic properties. In this study we have utilized the muon spin rotation/relaxation (mu+SR) technique to study the magnetic properties of the MgReO4 compound. To the best of our knowledge, this is the first investigation reported on this interesting material, that is stabilized in a wolframite crystal structure using a special highpressure synthesis technique. Bulk magnetic studies show the onset of an antiferromagnetic (AF) long range order, or a possible singlet spin state at T-C1 approximate to 90 K, with a subtle second hightemperature transition at T-C2 approximate to 280 K. Both transitions are also confirmed by heat capacity (Cp) measurements. From our mu+SR measurements, it is clear that the sample enters an AF order below T-C1 = T-N approximate to 85 K. We find no evidence of magnetic signal above TN, which indicates that T-C2 is likely linked to a structural transition. Further, via sensitive zero field (ZF) mu(+) SR measurements we find evidence of a spin reorientation at T-Cant approximate to 65 K. This points towards a transition from a collinear AF into a canted AF order at low temperature, which is proposed to be driven by competing magnetic interactions.