Recent advances in active materials and fabrication techniques have enabled the production of cyclically self-deployable metamaterials with an expanded functionality space. However, designing metamaterials that possess continuously tunable mechanical properties after self-deployment remains a challenge, notwithstanding its importance. Inspired by push puppets, we introduce an efficient design strategy to create reversibly self-deployable metamaterials with continuously tunable post-deployment stiffness and damping. Our metamaterial comprises contracting actuators threaded through beads with matching conical concavo-convex interfaces in networked chains. The slack network conforms to arbitrary shapes, but when actuated, it self-assembles into a preprogrammed configuration with beads gathered together. Further contraction of the actuators can dynamically tune the assembly's mechanical properties through the beads' particle jamming, while maintaining the overall structure with minimal change. We show that, after deployment, such metamaterials exhibit pronounced tunability in bending-dominated configurations: they can become more than 35 times stiffer and change their damping capability by over 50%. Through systematic analysis, we find that the beads'conical angle can introduce geometric nonlinearity, which has a major effect on the self-deployability and tunability of the metamaterial. Our work provides routes towards reversibly self-deployable, lightweight, and tunable metamaterials, with potential applications in soft robotics, reconfigurable architectures, and space engineering.