This paper studies the libration-free cargo transfer control of a partial space elevator where the main satellite may change its orbital state in the transfer period. The orbital motion of the main satellite in the climber transfer period is first studied. Then, a reduced-order libration-free dynamics of the partial space elevator is derived. Accordingly, a novel libration-free switching control strategy is proposed to stabilize the cargo transportation with two alternating controllers. The Controller I controls the cargo speed in the libration-free mode by a shrinking horizon model predictive control based on the reduced-order libration-free dynamic mode of the partial space elevator. This leads to high computational efficiency in control. Once the libration is induced by the cargo transfer, the control turns off the Controller I and activates the Controller II to suppress the libration to zero within one time step of the Controller I by a novel prescribed-time control law based on the fixed-time control scheme. The stability of the control is proved in the Lyapunov framework. The validity and effectiveness of the proposed control strategy are demonstrated by computation simulation. Simulation results reveal that the proposed control strategy is effective in keeping stable cargo transportation while ensuring the equilibrium state at the end of transportation.