During the processing of covered yarn, the stability of the tension of the yarn unwinding from the package directly affects the quality of the covered yarn. In this work, a mathematical model of the yarn unwinding balloon was established, and the influence of Coriolis force and air resistance on the balloon was considered. Further, the balloon shape of the yarn at different unwinding linear speeds was studied. An experimental observation platform for the yarn balloon morphology was built. The comparative analysis of the balloon shape obtained using the experimental observation and the theoretical research results showed that the mathematical model of the yarn unwinding balloon is correct. The phenomenon of scratching the upper edge of the package during the unwinding of spandex should be avoided. Finally, in order to maximize the diameter of the spandex package, the taper of the core yarn bobbin and the minimum yarn guide distance was determined based on the guidance of the theoretical simulations and experimental analysis. The research results provide a design base for the layout and structural dimensions of the follow-up elastic fiber passive precision tension control device.