We report on an investigation of the peculiar electronic structure of the layered semiconductor InSe by magneto-optical experiments under high pressure up to 5 GPa. Magneto-absorption spectroscopy is performed under pulsed magnetic field up to 53 T using a specific setup. Excitonic magnetofingerprints and high-field oscillatory magnetoabsorption yield significant details on the band structure. In addition, the application of an external pressure unveils phenomena that confirm the specific k . p model proposed for this compound on the basis of earlier measurements.