The schematic shows each end of transformer secondary connected to a diode
....a filter cap across it ...and a transistor with a zener diode on the base at
the output of the power supply. Can I modify this with something to get rid of
the hum?

Hmm... Possibly a capacitor across the output might help. Or even replacing the
existing capacitor. It may have derated with age or, because there's a zener
involved that may keep the ripple down, even opened.

Any corroboration from others? G

Bill, K5BY