The existence of the hole on a plate affects the calculation of eddy current problems. Consequently, the accuracy for the prediction of the material properties decreases if the effect of the hole is not taken into account. In this paper, a novel analytical method based on the modified integration range is proposed which can address the presence of the hole. Due to the presence of the hole, the conventional Dodd-Deeds analytical solution cannot be used to calculate the inductance change. Therefore, a revised upper integration limit is introduced to replace the original limit -- ∞ when using the co-axially air-core electromagnetic sensor. With the presence of the hole, the magnitude of the received signal reduces, and the peak frequency feature changes. The analytical method is validated by measured and numerical simulation results. It is found that the upper limit is related to the radius of the open hole. With the new technique, the thickness of sample plates with holes can be estimated based on the peak frequency feature.