This paper provides an efficient and accurate hybrid method to price American standard options in certain jump-diffusion models and American barrier-type options under the Black–Scholes framework. Our method generalizes the quadratic approximation scheme of Barone-Adesi and Whaley and several of its extensions. Using perturbative arguments, we decompose the early exercise pricing problem into subproblems of different orders and solve these subproblems successively. The solutions obtained are combined to recover approximations to the original pricing problem of multiple orders, with the zeroth-order version matching the general Barone-Adesi–Whaley ansatz. We test the accuracy and efficiency of the approximations via numerical simulations. The results show a clear dominance of higher-order approximations over their respective zeroth-order versions and reveal that significantly more pricing accuracy can be obtained by relying on approximations of the first few orders. In addition, they suggest that increasing the order of any approximation by one generally refines the pricing precision; however, this happens at the expense of greater computational costs.