The improved averaging method suggested in our previous report is developed in order to obtain a highly accurate periodic solution for a strongly nonlinear system. In this reports, the Jacobian elliptic sine and delta functions are applied as generating solutions for harmonically excited Duffing-type equations with softening spring and snap through spring, respectively. Two types of stability criterion for approximate solutions obtained by the present method are also discussed. The present method is applied to the systems with some kind of nonlinear spring. By comparing with the very accurate numerical solutions obtained by applying the shooting method, it is confirmed that the present method gives more accurate solution and more accurate results for stability of the solution than those obtained by the traditional averaging method using the trigonometric function as a generating solution.