We consider an uplink power allocation scheme for single-carrier frequency-division multiple access (FDMA) with iterative multiuser detection, called single-carrier grouped FDMA (SC-GFDMA). SC-GFDMA is a non-orthogonal scheme in which several users share a single time-frequency resource. Hence, the uplink signal of a user can be regarded as both a signal and a source of interference. The signal power of each user should be sufficiently high to ensure reliable signal detection and sufficiently low to suppress inter-user interference. That is, the transmit power of each user should be adjusted appropriately to achieve high spectral efficiency. In this context, a power control method for an uplink SC-GFDMA system is proposed by analyzing the signal-to-interference-plus-noise ratios of users sharing each time-frequency resource. In particular, the uplink spectral efficiency is improved by limiting the transmit power of each user according to a criterion derived using a semi-analytic method called signal-to-noise ratio-variance density evolution. Simulation results demonstrate that the proposed method can significantly increase the spectral efficiency of the system, even with a considerably reduced total transmit power.