This paper investigates the stochastic model predictive control (SMPC) problem with polyhedral constraints for linear discrete-time systems subject to additive stochastic disturbances. To address the non-convexity of the chance constraints in the original optimization problem, Cantelli''s inequality and variable substitution techniques are employed to transform them into second-order cone (SOC) constraints. Combined with the convex quadratic reformulation of the cost function and terminal constraints, an approximate convexification of the closed-loop optimal control problem is achieved. On this basis, an active terminal covariance allocation strategy is proposed. By parameterizing the state feedback gain and terminal covariance, this strategy enables the explicit regulation of the system state distribution. This effectively overcomes the implicit coupling between dynamic performance and stability inherent in traditional designs, thereby significantly improving the closed-loop performance. Furthermore, to handle additive unbounded disturbances, a theoretical guarantee of recursive feasibility is established based on the nominal state evolution. An expected upper bound on the control cost is also derived, theoretically ensuring the stability of the closed-loop system under stochastic disturbances. Finally, theoretical analysis and numerical simulations verify the superiority of the proposed method in strictly satisfying chance constraints and reducing control conservatism, providing a systematic and efficient design framework for the MPC of stochastic systems.