Robotic arms built from stiffness-adjustable, continuously bending segments serially connected with revolute joints have the ability to change their mechanical architecture and workspace, thus allowing high flexibility and adaptation to different tasks with less than six degrees of freedom, a concept that we call malleable robots. Known stiffening mechanisms may be used to implement suitable links for these novel robotic manipulators; however, these solutions usually show a reduced performance when bending due to structural deformation. By including an inner support structure this deformation can be minimised, resulting in an increased stiffening performance. This paper presents a new multi-material spine-inspired flexible structure for providing support in stiffness-controllable layer-jamming-based robotic links of large diameter. The proposed spine mechanism is highly movable with type and range of motions that match those of a robotic link using solely layer jamming, whilst maintaining a hollow and light structure. The mechanics and design of the flexible spine are explored, and a prototype of a link utilising it is developed and compared with limb segments based on granular jamming and layer jamming without support structure. Results of experiments verify the advantages of the proposed design, demonstrating that it maintains a constant central diameter across bending angles and presents an improvement of more than 203% of resisting force at 180 degrees.