Recent photometric surveys of Trans-Neptunian Objects (TNOs) have revealed that the cold classical TNOs have distinct color characteristics in z-band, and occupy a surface class separate from those of the dynamically hot TNOs. This finding also suggested the presence of an absorption band in the reflectance spectra of cold classicals at wavelengths above 0.8 micron. Here we present reflectance spectra spanning 0.55-1.0 micron for six TNOs occupying cold classical orbital parameter space. Five of our spectra show a clear and broadly consistent flattening behaviour above 0.8 micron that diverges from their linear red optical continuum; this finding agrees with trends outlined by photometric colour data. Our sixth spectrum remains nearly linear across the full range observed, however, indicating that it likely originated alongside the hot TNO population, and is currently interloping in the cold classical dynamical range. Despite predictions, we find no evidence that the spectral flattening is caused by an absorption band centered near 1.0 micron. Instead, we predict that the, presumably, organic materials on the surfaces of the cold classical TNOs are chemically and structurally different to those on the surfaces of the dynamically hot TNOs, and therefore exhibit different spectral behaviour. This plausibly results from differences in the conditions under which the two populations formed, and/or variation in the extent or type of surface evolutionary process experienced by objects in each population.