Serpentine structures are of growing interest due to its unique mechanical and physical properties for applications in stretchable electronics, mechanical sensing, and biomedical devices. Mechanics-guided, deterministic three-dimensional (3D) assembly provides routes to form remarkable 3D structures, which in turn significantly improve its potential for applications. Therefore, an accurate postbuckling analysis is essential to the complex 3D serpentine structures with arbitrary geometry/material parameters. Here, simple, analytical expressions are obtained for the displacement and effective rigidity of serpentine structures during postbuckling. By tuning geometry parameters, the amplitude of assembled 3D serpentine structures can span a very broad range from zero to that of a straight ribbon. The analytical model can be used in design, fabrication, and application of versatile 3D serpentine structures to ensure their compatibility with the ultra-low rigidity biological tissues. A hierarchical 3D serpentine structure with ultra-low rigidity is presented to demonstrate the application of the analytical model.