The inverse protein folding problem is that of designing an amino acid sequence which has a prescribed native protein fold. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. Previously, tubular structures for a three-dimensional (3D) hexagonal prism lattice were introduced and their stability was formally proved for simple instances under the hydrophobic-polar (HP) model of Dill. In this article, we generalize the design of tubular structures to allow for much larger variety of designable structures by allowing branching of tubes. Our generalized design could be used to roughly approximate given 3D shapes in the considered lattice. Although the generalized tubular structures are not stable under the HP model, we can prove that a simple instance of generalized tubular structures is structurally stable (all native folds have the designed shape) under a refined version of the HP model, called the HPC model. We conjecture that there is a way to choose which hydrophobic monomers are cysteines in all generalized tubular structures such that the designed proteins are structurally stable under the HPC model.