A stochastic oscillatory integral is a probabilistic counterpart to a Feynman path integral, and the stationary phase method for stochastic oscillatory integrals relates to the semi-classical limits in the Feynman path integral theory. A complete stationary phase method for stochastic oscillatory integrals has not been achieved yet. In this paper, a new class of amplitude functions of stochastic oscillatory integrals is introduced, and it is shown that the stationary phase method works for the class.