The cylindrical vector vortex beam presents a cylindrically symmetrical polarization state distribution and has a helical phase wavefront, which can be described by a hybrid-order Poincaré sphere. In this paper, based on the deduction of Jones matrix, we propose and demonstrate a method for generating arbitrary cylindrical vector vortex beams on a hybrid-order Poincaré sphere using vortex half-wave plates. By using the combination of vortex half-wave plate, half-wave plate, and quarter-wave plate, it is feasible to generate cylindrical vector vortex beams at different positions on the hybrid-order Poincaré sphere. The range of orders of hybrid-order Poincaré sphere beams can be further expanded by cascading multiple vortex half-wave plates. In experiments, we have measured the polarization state distribution of various beams at different positions on the hybrid-order Poincaré sphere, and verified their polarization order and topological charge. This simple and flexible method can realize the independent manipulation of topological charge and polarization order, and has potential value for applications including optical micro-manipulation, optical detection and sensing, and design of novel vector vortex laser sources.