To evaluate the seismic safety of components in a structure, an in-structure response spectrum (ISRS) must be obtained, and this also applies to seismically isolated structures. The main variables for designing seismic isolators are effective stiffness and effective damping, which can be given as the characteristic strength and secondary stiffness in seismic isolators for nonlinear behaviors. Many studies on the ISRS of isolated structures have been conducted to evaluate the effects of these two variables of isolators, but the effect of other variables related to the hysteresis curve of isolators also needs to be studied. This study focused on the effect of the initial stiffness of isolators on an ISRS because there were no clear criteria for determining the initial stiffness in isolator design standards, which has an effect on the ISRS that cannot be ignored. As a result, the initial stiffness contributed significantly to the ISRS not only at the natural frequency of the structure but also at low frequencies. An analysis was also performed in terms of the uncertainty of each variable. The sharpness of the yield point in the hysteresis curve was implemented using the Bouc–Wen model, and its impact was analyzed. In addition, the frequency content of the ISRS depending on the seismic intensity, which can considerably change the nonlinear hysteresis behavior, was examined. Through this study, although secondary stiffness and characteristic strength are the most important characteristics of seismic isolation design, it was confirmed that other variables also have a significant impact on the frequency content of an ISRS. Based on this study, the considerations when developing the ISRS of an isolated structure can be established.