Your circuit will work fine, and as it stands the two DC supplies will not interfere with each other. Of course, you must ensure that any power you derive from your additional supply does not overload the transformer. Any current you draw from your second supply is increasing the current through the transformer, over and above the amount that the first supply drew on its own.
The biggest issue you face may have to do with how the systems you connect to those two sources interact. You now have two ground points, the original labelled '0V', and your new one with a ground symbol, bottom right.
These are not the same, and by measuring the AC voltage between them, you will find a significant potential difference.
Consequently, any signal derived from a circuit connected to your new second supply will be relative its own particular ground, and in relation to the other supply's ground, it will look like a mess of noise and AC. You cannot solve this problem by connecting the two grounds to each other, because that will cause diodes to be short-circuited, and smoke to happen.
In other words, this arrangement will only be of any use to you if there will be absolutely no communication (or any kind of connection) between the circuitry on each DC supply. If all you want is to power a completely independent system from the second DC source, you are good to go. Otherwise you can expect serious complications.