Step Down Transformer Wire Diameter Theory

When you use a transformer for stepping down voltage and stepping up current, you need many turns on the primary side of the transformer and less turns on the secondary side.

If you wind a second coil on the output side and connect it in parallel with the previous secondary coil, the output current should be higher than before. This implies that thicker wire on the secondary yields a higher output current.

I think that if the primary coil and secondary coil have the same wire gages then the voltage on the secondary will be stepped down, but current wont go up.