Foldable structures have been of great interest due to their ability to reduce in size from deployed to folded state, enabling easier storage in scenarios with space constraints such as aerospace and medical applications. Hexagonal structural components have been of interest, due to their ability to tessellate, or cover without gap, 2D and 3D surfaces. However, the study on effective folding strategies for hexagon-based structures and the hexagon geometry itself is limited. Here, we report a strategy of snap-folding hexagonal rings, to result in folded states with only 10.6% the initial area of a single ring. Motivated by this significant packing, we utilize a combination of experiments and finite element analysis to study effective folding strategies and packing abilities of various 2D and 3D hexagonal ring assemblies, with structures that can be folded to 1.5% and 0.4% of their initial area and volume, respectively. The effect of geometric parameters of hexagonal rings on the mechanical stability of their assemblies is investigated. Additionally, the instabilities of rings can be utilized to facilitate the automatic deployment of folded ring assemblies under small perturbations. Furthermore, an assembly with rigid functional panels is explored to demonstrate the functionality and design space for hexagonal ring assemblies. With such significant demonstrated area and volume changes upon snap-folding, it is anticipated that hexagonal ring assemblies could inspire future aerospace or biomedical designs, where reconfiguration and large packing are required.