In this paper, we propose a hybrid second-order method for homogenous polynomial optimization over the unit sphere in which the new iterate is generated by employing the second-order information of the objective function. To guarantee the convergence, we recall the shifted power method when the second-order method does not make an improvement to the objective function. As the Hessian of the objective function can easily be computed and no line search is involved in the second-order iterative step, the method is not time-consuming. Further, the new iterate is generated in a relatively larger region and thus the global maximum can be likely obtained. The given numerical experiments show the efficiency of the proposed method.