You will need a 20-0-20 transformer to produce ~ +/- 30v full wave dc with a capacitive filter.
Assuming your transformer is rated for 3A, you will need to remove half of the turns on each leg of the secondary. Therefore if it is a single, or split bobbin type core, unless they have devised some sneaky way of putting the windings on that has escaped me the last 30+ years, this means you will have to remove completely one winding and half of the first one, make a new centre tap, and wind on the correct number of turns to complete the second winding again. If it doesn't buzz under load, I'll be surprised, because I doubt you will be able to hand wind it with even tension from start to finish. Although a good dip in transformer varnish works wonders.
It would be far easier, and probably much safer for you, to just buy a 20-0-20 3A transformer. Remember one nick in the enamel between turns and the transformer goes up in smoke...
If you have to re-wind a transformer, never be tempted to use second hand enamelled wire. Always replace the windings completely with new wire.
Never ever join wire inside the transformer winding, always bring the tails out neatly and make any joint outside of the winding window.
Always keep correct and even tension on the wire as you wind, any slack or sloppy winding that creates an anomaly will only be amplified in later layers, possibly resulting in lack of window space for the windings.
Always properly insulate your windings when complete, including a final tape layer, this will help with mechanical strength too.