In this paper, we characterize, based on stochastic geometry, the uplink coverage probability with a unified power control scheme built upon realistic path loss models and user equipment (UE) constrained transmit power. To improve their uplink connectivity, active UEs are next assumed to move in a random direction without prior knowledge of their nearest base station location, namely the blind cell search (BCS) movement. A tractable expression of the uplink handoff rate is then derived and the induced uplink coverage probability following the BCS movement is evaluated. The results show different echoes of the uplink coverage probability depending on the serving UE profile (stationary or mobile) and the considered path loss model, which suggests new insights into the design of uplink system parameters.