[I can't put phone numbers on this post ]
Due to the recent privacy laws in Australia which have now enforced since the 1990s one must contact the State Government agencies..... you need to do a FOI and they may not give information as it may have been destroyed after 100 years. Since you talk about the 20/30s you may have better chance of viewing the records at the agency office or they send extracts....
since you do not specify race of person , which in the case of aborigines, there are different rules with the alligned agency. Have a talk firstly to the local registrars office to get the correct government phone number as the switchboard will take too long.
Since the period you seek is a little aged, the Barnardos agency may not have data but worth having a chat to. Its a little early for a Red Cross file and is in a period of relative peacetime so contacting them on this is a waste of time.
If you do suceed in getting a name ..... I would do some serious newspaper hunting or acquire death certificate of adopted persons mother...... good luck