Soft manipulators, a relatively novel class of robots, are increasingly prevalent in the field of robotics. Due to the material nonlinearity and the absence of links and joints which are typically used to derive their kinematics and dynamics, the exact modeling and effective control frameworks of such soft robots are much more difficult than the traditional rigid counterparts. To achieve global control of the end-effector position of the soft manipulator actuated by cables, this work proposes a hybrid control strategy by unifying the piecewise linear strain (PLS) Cosserat model and radial basis function neural network (RBFNN) to approximate the Jacobian matrix of any end-effector position in the whole workspace, and then designs a global control scheme based on the approximation of Jacobian matrix. The theoretical convergence proof and experimental validation are provided for the designed global controller. The results corroborate that the proposed control strategy has excellent accuracy and robustness in tracking desired time-varying trajectories through different experiments.