For one, you need to put a load on the supply. The load will drop across the two 47 ohm resistors. At 20 mA, you won't drop much.
One way is to add a series resistor, another is the regulator I suggested and yet a third way is to use another transformer in series with the primary to drop the secondary voltage or two transformers to drop the secondary voltage because of the center tap. You could use a Bifilar wound torroid transformer though. e.g 2x18 V secondaries.
You basically have to phase them properly. A series connected transformer will either add or subtract. So, it's basically connect both primaries in parallel. Connect the secondaries in series. Measure the combined AC voltage. If too high, reverse either the primary or secondary. Then connect back to your circuit. Probably a little tough to visualize, but you end up applying a smaller voltage to the primary of the existing transformer. This really won;t work because of the 6.3 V winding. To reduce the secondary, you need two transformers. So, it's an OOPs.