Can new cable be run from switch to ceiling?
As @Tyson mentions in the comments there are devices listed for repair of NM cable such as the Tyco device mentioned in this answer.  However it would actually be pretty hard to get that in there, and I believe it's only listed for repair, not for extending.
You can mount a junction box and use that to house a splice to extend the cable.
Inacessible junction boxes are not permitted by code (see below.)  However this box would be accessible by removing the fan.  By most interpretations, that does not count as  **removing part of the building. **
It looks like there is room to mount a handy box out of the way of the fan and the existing cabling.  If not, you may be forced to remove the drywall.

314.29 Boxes, Conduit Bodies, and Handhole Enclosures to Be Accessible.
Boxes, conduit bodies, and handhole
enclosures shall be installed so that the wiring contained
in them can be rendered accessible without removing any part of the building or, in underground circuits, without
excavating sidewalks, paving, earth, or other substance that
is to be used to establish the finished grade