Measurements in field-effect transistors have indicated that the Boltzmann thermal limit of the subthreshold swing, (k B T /q) ln 10, is not followed down to deep-cryogenic temperatures. Instead, there is a nonlinear deviation from this limit which cannot be explained with the present theory. In this paper, we derive a new theory which resolves this discrepancy by including the source-drain tunneling through a band tail. We demonstrate that the additional tunneling component of the current becomes dominant over the diffusion current at a sufficiently low cryogenic temperature. A band tail in the electrostatics, with only diffusion transport, does not explain the excess subthreshold swing at deep-cryogenic temperatures, neither does only self-heating, nor does a nonuniform density of interface traps in the bandgap with Fermi-Dirac occupation. The proposed theory is experimentally validated with our measurements in advanced CMOS technology down to 4.2 K. Finally, we assess the obtained densities of interface traps in the presence of band-tail tunneling.