The 28-year-old had been reading a self-help book promoting “rediscovery by dumping your past”.
Her family say it is possible she has been helped to disappear in Australia by a spiritualist group or cult.
Melody, from Bolton, disappeared two weeks ago from a friend’s house in Sydney where she had stayed the night. She has not been seen since and has not been in contact with any family or friends.
Her father, Hugh, and brother Kevin flew to Australia last week to help in the search for Melody, a former Bolton School pupil. After speaking to some of her friends, they believe she may have decided to follow the teachings of a self-help book called The Power of Now.
Written by German Eckhart Tolle, the book suggests people can find inner-fulfilment by forgetting their past.
Her father said: “The book is about dumping your past to come to terms with your problems.
“Apparently she has been promoting it to her friends and we think there is a possibility she has decided to discover her spiritual side.”
Mr O’Gara, from Breightmet, said there were groups or cults in Australia who followed the same belief.
“They could help someone disappear.
“If they have, without endangering her seclusion, we would urge people to tell the police just so that we know she is alive so we can end all this.
“We would not pursue her. The relief of that one call that can verify she is alive would be tremendous.”
Police in Kings Cross, Australia, have confirmed the theory to be one of a number they are investigating.
There is also the possibility she committed suicide after a man reported seeing a woman fitting Miss O’Gara’s description sitting at a cliff edge.
Her bag, which contained her purse, was later found at the scene.
However, her father believes there is little to suggest the intelligent woman who had a wide circle of friends would kill herself.
Mr O’Gara said: “She has never indicated anything other than the fact that she was enjoying herself on her holiday.”
“We have been through her diary and there is nothing in there at all about her mood or feelings.”
Mr O’Gara and his son are frantically trying everything they can to find her and have no plans to return home soon.
“This is the worst experience of my life — you just cannot imagine it.”