Divers have been blasting holes inside the stricken steel-hulled liner to help them look for those unaccounted for.
The disaster earlier this month has so far claimed 16 lives with at least another 16 people missing.
On Tuesday, the body of a woman was discovered after divers managed to enter a submerged section on the third floor deck of the Concordia.
The victim was found with a life jacket and her body was taken to a mortuary on the mainland.
The ship, carrying more than 4,000 passengers and crew, ran aground and capsized on January 13 after the captain veered from his approved course.
Work on pumping out 500,000 gallons of heavy duty diesel from the wreckage of the stricken vessel has now begun.
The £5m operation to remove the fuel from the ship's 13 double bottomed tanks and ten single bottom tanks is expected to last between four and six weeks.
The captain, Francesco Schettino, is currently under house arrest after he steered his 114,000-ton ship onto rocks close to the island of Giglio allegedly in a ''sail by'' salute to impress a crew member.
His wife, Fabiola Russo, has defended her husband, saying: "He is not a monster" - and she spoke out as calls increased for prosecutors to widen their probe into the disaster.
In an interview with glossy Italian weekly magazine Oggi, Fabiola said there was a witch-hunt against her husband, who is under investigation for multiple manslaughter, abandoning ship and causing a ship wreck.
Fabiola, who lives with Schettino and their 17-year-old daughter in Meta di Sorrento, near Naples, said: "My husband is not a monster. He was always the go-to guy for the crew.
"I am the wife of a seaman, so I am used to doing things on my own. I tackle situations and I resolve them.
"This is a witch-hunt, everyone is out to get my husband. The media has made out he is to blame. He loves ships, he knows all about them and has never stopped studying them, to learn what they are capable of and what their limits are.
"He didn't stop at just sailing ships. That's why for his crew he was always the go-to man, a maestro. He is decisive, stable and lucid, he analyses situations, understands them and then acts.
"At times Francesco seems arrogant because he puts himself above the people he is speaking to and often tells them to shut up. He wasn't in charge of the Concordia by chance, he knows how to do his job but sometimes even those who know how to do their job can make mistakes - that's if he did make a mistake."
Fabiola also revealed how they had once been fined for sailing too close to the shore in their sailing boat and added: "What we really like is canoeing and to row together you need to be in harmony and my husband and I are."
She spoke out as Beniamino Deidda, the prosecutor general of Tuscany, the Italian region Giglio is in, called for attention in the investigation to also focus on the Concordia's owners Costa Cruises.
He said: "For the time being the main focus of attention is the captain, who, it has been shown, was tragically inadequate. But who chose him?
"The lifeboats did not lower properly, the crew didn't know what to do, they were unprepared in dealing with an emergency, wrong orders were given, such as being told to return to cabins.
"The confusion that has emerged shows that there was an incredible lack of attention to safety procedures. Safety should have been rehearsed beforehand. You cannot blame all of this carelessness on the conduct of the captain. That's why this investigation cannot exclude others."