Significance The thermodynamics of glasses (disordered solids) exhibits experimentally a universal behavior upon cooling below 10 K that is very different from regular solids. Theoretical interpretations of this discrepancy are notably hard to vindicate from first principles. Here, through the analytic solution of a simple microscopic model of a glass, we propose an alternative scenario to quantum tunneling between similar energy states. We suggest that the anomalous scaling gets broader in temperature range as the system is uncompressed toward the jamming transition where, classically, soft spherical particles would be in contact in their ground state. This scaling is thus most easily detected close to jamming and fades out away from it.