Gas storage reservoirs are crucial infrastructures for strategic reserves, emergency peak regulation, and safeguarding national energy security. However,the complex injection and production processes and frequent pressure fluctuations pose severe challenges to safe production. The through-tubing retrievable packer, as a key downhole equipment for through-tubing operations, has the advantages of high efficiency, low risk, and simple operation. It shows the great application potentialities in the fields of temporary plugging and seal verification of immobile pipe strings. However, the through-tubing packer with the traditional laminated steel sheet structure cannot meet the on-site requirements for the super-expansion sealing of using 4-1/2 tubing to seal the 7-inch casing in gas storage reservoirs. It is urgent to solve the problem of sealing a large-diameter casing with a small-diameter tubing while taking into account the sealing performance and strength requirements of the rubber element. Here, this paper innovatively put forward a design scheme for a new corrugated-skeleton through-tubing retrievable packer. It determined the size parameters and material selection of the corrugated skeleton, three-rubber-element structure and other key components. By establishing the setting mechanical models of through-tubing packer, the setting mechanism was elucidated and the effective setting conditions for downhole operations were defined. The 3rd-order Ogden model was selected to conduct a hyperelastic constitutive analysis on the polyurethane rubber element, and the ABAQUS finite element simulation software was adopted to simulate the super-expansion process of the packer. The results indicated that the overall complete super expansion could be achieved within 30 seconds, verifying the feasibility of the super expansion behavior. On this basis, the geometric size parameters of the corrugated skeleton and sealing rubber element of the super-expansion corrugated-skeleton through-tubing retrievable packer were further optimized. The simulation results showed that when the profile expression of the corrugated skeleton is y=3sin1.5x and the outer diameter of the outer rubber element is 84 mm, the overall stress change of the packer is smooth, the setting performance is good, and thus the stress concentration can be effectively reduced, and the sealing performance and setting effect can be improved. This super expansion corrugated-skeleton through-tubing retrievable packer can meet the pressure-bearing requirements of the complex working conditions of gas storage reservoirs, reduce the stress concentration of the sealing rubber element, extend the service life of the downhole tool, and provide theoretical support and design reference for the design of super expansion through-tubing retrievable packers in gas storage reservoirs.