Like George Bush running against Al Gore, I'm going to have to withdraw my admission of defeat. After looking into it more, it seems this issue is more nuanced.
My undergrad orgo book (Bruice) doesn't touch this exact issue of chloride-carbon bonds, but it does state that for all of the hydrogen halides the sigma bond is formed from the H's s orbital and the halogen's sp3 orbital. The argument is that although the bond angle here doesn't give us any direct info regarding the orbitals involved in bonding (since it's obviously linear), it is best to consider the halogen as sp3 since the electron repulsion would be minimized.
Furthermore, from Grossman's "The Art of Writing Reasonable Organic Reaction Mechanisms" he states:
"The hybridization of an atom is determined as follows. Hybrid orbitals are used to make sigma bonds and to hold lone pairs not used in resonance; p orbitals are used to make pi bonds and to hold lone pairs used in resonance, and they are used as empty orbitals. To determine the hybridization of an atom, add up the number of lone pairs not used in resonance and the number of sigma bonds (i.e. atoms to which it is bound). If the sum is four, the atom is sp3-hybridized. If the sum is three, it is sp2-hybridized. If the sum is two, it is sp-hybridized"
Following Grossman's rule, the chloride should be considered sp3. Indeed several other websites and homework sets online agree that chlorine is sp3. However there are also textbooks that say in Cl2 the bond is formed via the 2p orbitals.