The hypercompressor is one of the core facilities in low density polyethylene production, with a discharge pressure of approximately 300 MPa. A packing cup is the basic unit of cylinder packing, assembled by the interference fit between an inner cup and an outer cup. Because the shrink-fitting prestresses the packing cup, serious design is needed to gain a favorable stress state, for example, a tri-axial compressive stress state. The traditional method of designing the interference fit value for packing cups depends on the shrink-fit theory for thick-walled cylinder subject to internal and external pressure. According to the traditional method, critical points are at the inner radii of the inner and external cup. In this study, the finite element method (FEM) has been implemented to determine a more accurate stress level of packing cups. Different critical points have been found at the edge of lapped sealing surfaces between two adjacent packing cups. The maximum Von Mises equivalent stress in a packing cup increases after a decline with the rise of the interference fit value. The maximum equivalent stress initially occurs at the bore of the inner cup, then at the edge of lapped mating surfaces, and finally at the bore of the outer cup, as the interference radius increases. The traditional method neglects the influence of axial preloading on the interference mating pressure. As a result, it predicts a lower equivalent stress at the bore of the external cup. A higher interference fit value accepted by the traditional method may not be feasible as it might already make packing cups yield at the edge of mating surfaces or the bore of the external cup. Along with fatigue analysis, the feasible range of interference fit value has been modified by utilizing FEM. The modified range tends to be narrower and safer than the one derived from the traditional method, after getting rid of shrink-fit values that could result in yielding in a real packing cup.