We report the results of systematic measurements of the group velocity dispersion (GVD) in chalcogenide glass (ChG) bulk samples, composite ChG fibers, and robust high-index-contrast nanotapers. The composite ChGpolymer fibers are drawn from an extruded multimaterial preform incorporating a thick built-in polymer jacket that is thermally compatible with the ChG used, and the nanotapers are then produced without removing the polymer. We isolate the contributions of material and waveguide GVD to the total dispersion in the nanotapers and support the results with finite-element simulations. These results indicate many possibilities for dispersion engineering and nonlinearity enhancement in all-solid index-guiding ChG fibers stemming from the flexibility of this fiber fabrication methodology.