There are a couple of errors in my circuit.
There should be a diode in reverse parallel with the relay coil.
Another thing I've overlooked is that it could be oscillating because the filter capacitor is being discharged when the coil is activated causing the output voltage to drop slightly resulting in it being deactivated.
I can think of two solutions:
Add a capacitor from the transistor's base to 0V. This might help keep the transistor on for a long enough length of time for the relay to connect to the higher voltage tap.
Add a high value resistor (try 100k to 1M) from the + side of the rectifier to the transistor's base. This should add some hysteresis so when the relay does engage the transistor will turn on even more.
Failing that, are you sure the filter capacitor is large enough?
What's the secondary voltage of your trasformer?
I would recommend using the solid state version: it should be much more reliable.
You might have to tweak a few component values, remember I haven't tested or simulated any of these circuits.