The Bayer process is the main method of alumina production worldwide. The use of low-quality bauxites for alumina production results in the formation of a significant amount of technogenic waste—bauxite residue (BR). The Bayer reductive method is one possible way to eliminate BR stockpiling, but it requires high-pressure leaching at temperatures higher than 220 °C. In this research, the possibility of boehmitic bauxite atmospheric pressure leaching at both the first and second stages or high-pressure leaching at the second stage with the simultaneous reduction of hematite to magnetite was investigated. Bauxite and solid residue after NaOH leaching were characterized using XRD, SEM-EDS, and Mössbauer spectroscopy methods. The first stage of leaching under atmospheric pressure with the addition of Fe(II) species in a strong alkali solution (330–400 g L–1Na2O) resulted in a partial reduction of the iron minerals and an extraction of more than 60% of Si and 5–25% of Al (depending on caustic modulus of solution) after 1 h. The obtained desilicated bauxite was subjected to atmospheric leaching at 120 °C in a strong alkali solution (350 g L−1) or high-pressure leaching at 160–220 °C using the Bayer process mother liquor in order to obtain a concentrate with a magnetite content higher than 83 wt. %.