So long story short, despite the following bug ID saying that the Catalyst 2960-L switches are supported as of various versions, most of them are actually NOT supported still despite the BugID being marked as fixed.
https://bst.cloudapps.cisco.com/bugsearch/bug/CSCvf16332/?rfs=iqvred
However, TAC was willing to manually add the OID of my switches to my CER server. Basically all it required was an additional line in the devices.xml file on the server. After doing that, I can now use phone tracking on my 2960-L switches.
So, is there some particular reason why Cisco doesn't periodically release COP files that update the devices.xml file to add support for new switches? It seems overkill to have to upgrade an entire CER server version just to get support for newer switches, when it seems that replacing the devices.xml file with a new one is all that is required.
