We present ∼115 new spectroscopically identified members of the GD-1 tidal stream observed with the 5000-fiber Dark Energy Spectroscopic Instrument (DESI). We confirm the existence of a "cocoon," which is a broad (FWHM ∼ 2 932 ∼ 460 pc) and kinematically hot (velocity dispersion, σ ∼ 5–8 km s‑1) component that surrounds a narrower (FWHM ∼ 0 353 ∼ 55 pc) and colder [(σ = 3.09 ± 0.76 km s‑1)] thin stream component (based on a median per star velocity precision of 2.7 km s‑1). The cocoon extends over at least a 30∘ segment of the stream observed by DESI. The thin and cocoon components have similar mean values of [Fe/H]: ‑2.54 ± 0.04 dex and ‑2.47 ± 0.06 dex, suggestive of a common origin. The data are consistent with the following scenarios for the origin of the cocoon. The progenitor of the GD-1 stream was an accreted globular cluster (GC) and: (a) the cocoon was produced by pre-accretion tidal stripping of the GC while it was still inside its parent dwarf galaxy; (b) the cocoon comprises debris from the parent dwarf galaxy; (c) an initially thin GC tidal stream was heated by impacts from dark subhalos in the Milky Way; (d) an initially thin GC stream was heated by a massive Sagittarius dwarf galaxy; or a combination of some of these. Future DESI spectroscopy and detailed modeling may enable us to distinguish between these possible origins.