In the thinly bedded rock salt in China, the horizontal multi-stage leaching method can solve the problem of constructing an enormous salt cavern in a limited-thickness salt layer. This study aims to investigate how the final morphology of the salt cavern is affected by critical parameters of the horizontal multi-stage leaching method in different geological conditions, such as the injection rate of freshwater, tubing withdrawal distance, and the presence of an air cushion. These parameters will directly affect the shape of the salt cavern and thus influence the stability of the underground energy storage during operation. We developed an experimental device to simulate salt cavern construction using the horizontal multi-stage leaching method. By varying the parameters in the physical simulation experiments, horizontal salt cavern models with different morphologies and sizes were obtained. The experiments showed that the injection rate of freshwater significantly affects the salt cavern height, length, and maximum width. The tubing withdrawal distance mainly affects the shape of the roof of the salt cavern. Introducing an air cushion to protect the salt cavern roof during the construction of a salt cavern results in a “flat-top” morphology. According to these salt cavern models, three-dimensional geomechanical models for different geological conditions are established by using reverse engineering technology. After analyzing the stability of the salt cavern using these models and combining them with economic factors to recommend values for the critical parameters of horizontal multi-stage leaching method in different geological conditions are provided. This study provides reference value and a theoretical basis for the construction of horizontal salt cavern storage in China.