The rate of majority-carrier emission at the trap level located in the edge of the depletion layer is enhanced by the influence of a free-carrier tail. This causes a nonexponential change in transient capacitance with time after the application of a filling pulse. The emission rates of the midgap level in GaAs and the Au acceptor level in Si are measured by the constant capacitance technique which eliminates the additional nonexponential factor due to the high trap concentration. The apparent emission rate determined from the initial slope of transient voltage increases with decreasing the difference of the depletion widths in the emission and capture processes. The enhanced emission rate due to the free-carrier tail is observed even on the sample of the low carrier concentration. Fitting the calculated emission rate to the experimental results enables the estimation of the capture cross section of majority carriers in the trap level as well as the emission rate.