Yes, perfectly doable. The primary winding is almost universally buried under the existing secondary, so there are no particular safety isolation issues. I've done it several times for auxiliary windings.
Make sure you use normal insulated transformer wire, never bare uninsulated (obvious I know, but some have tried - and failed!). You'll want to do a test winding to establish the turns per volt and then wind accordingly, spreading turns evenly around to core. Teflon / Kapton / Polyester tape should be fine.
P.S. Try to find a little bobbin for the wire, small enough to fit through the center hole, trying to thread long lengths of un-tethered wire through the hole is a pain.