We investigate the tunability of the Goos–Hänchen shift experienced by a Gaussian light beam reflected from a multilayered heterostructure consisting of a nematic liquid-crystal cell sandwiched between electrodes and deposited on a magneto-electric/non-magnetic bilayer. Our calculations account for the optical anisotropy of the magnetic layer and of the liquid-crystal cell, as well as for the inhomogeneous refractive index distribution in the latter due to the reorientation of its molecules. We show that the Goos–Hänchen shift can be enhanced and controlled via the voltage applied to the liquid crystal cell and the magnetization direction of the magnetic film. We propose to exploit the voltage-induced tunability of the Goos–Hänchen shift in this system to design an optical sensor devoted to the detection of chemical vapors in the vicinity of the structure.