We studied the effect of physical pressure on the electronic and magnetic properties of ferrimagnetic double perovskites A2FeReO6 (A=Ca, Ba) using Re L2,3 edge x-ray absorption spectroscopy and powder diffraction measurements. Volume compression is shown to dramatically increase the magnetic coercivity (Hc) in polycrystalline samples of both compounds with ∆Hc/∆V ∼ 150-200 Oe/Å 3. A nearly eight-fold increase in Hc, from 0.2 T to 1.55 T, is obtained in Ba2FeReO6 at P=29 GPa. While no signs of structural phase transitions are seen in either sample to ∼ 30 GPa, the structural data points to a pressure-driven increase in tetragonal distortion of ReO6 octahedra. A sizable but pressure-independent Re orbital-to-spin magnetic moment ratio is observed, pointing to the critical role of spin-orbit interactions at Re sites. We present a j eff description of the electronic structure that combines effects of crystal field and spin-orbit coupling on the Re 5d 2 orbitals and use this description to provide insight into the pressure-induced enhancement of magnetic anisotropy.