GIGLIO, Italy (CNN) -

The complex operation to refloat the wrecked Costa Concordia cruise ship continued Tuesday -- but the family of the final missing victim still must wait several days for the vessel to be searched for his remains.

The last time anyone saw Russel Rebello alive, he was near the stern of the Costa Concordia, helping passengers into rescue boats.

But the 33-year-old waiter from India never escaped the doomed cruise ship. And he's the only victim of the 2012 accident whose remains haven't been found.

Officials told reporters Tuesday that the vessel would only be searched for Rebello's remains once it reaches its final destination, the Italian port of Genoa.

The seabed is being searched and those searches will intensify when the ship is gone, they said.

Salvage crews began the arduous task of refloating the ship on Monday morning so they can move it from its resting place off Giglio Island to Genoa to be dismantled.

On Tuesday, they began the process of lowering steel hollow boxes, or sponsons, which will be pumped full of compressed air to help lift the vessel further. Divers also began attaching more chains and cables to reinforce the bottom of the boat.

Once that's done, they'll raise the ship deck by deck, clearing any debris along the way.

After the ship is completely floated, a convoy of 10 boats will then begin a potentially treacherous journey. The vessel will be towed -- slowly and carefully -- 240 kilometers (150 miles) to Genoa, where it will be broken up.

It's been more than 2½ years since the ship ran aground off Italy's Giglio Island with more than 4,200 passengers aboard, killing 32 people in a disaster that drew global attention.

Cleanup job

Reporters were told at a briefing Tuesday that the salvage process was all under control.

Initial surveillance by remotely operated underwater vehicles showed no visible cracks or fissures on the bottom of the ship. This had been a concern because if repairs were needed, the operation would have been delayed.

Additional structural checks are now underway, with the salvage crews' efforts expected largely to focus over the next two days on the parts of the rotting hulk that remain under water.

The project's chief engineer, Franco Porcellacchia of Costa Crociere, the firm that owns the Concordia, said Tuesday it was most likely that the ship would start its final journey on Monday. Officials have said it will take five to six days for the vessel to reach Genoa once it gets going.

Porcellacchia also said the crews "have a commitment to clean the (sea) bottom to the best that we can." This will include the removal of sediment around the hulk and the underwater salvage platforms and cement bags used to support it when it was rotated to an upright position last year.

The crew will start to clean up once the ship is gone but are accepting tenders for removal companies to speed up the process. That bid will be awarded in August, with the removal process expected to take a year or more.

Giglio Mayor Sergio Ortelli said he was satisfied with the results so far of tests on water quality around the ship.

The decision to undertake the expensive and difficult process of refloating the Costa Concordia, rather than trying to dismantle it on site, was taken because of environmental concerns.

Italy's Environment Minister Gian Luca Galletti said Monday it was too risky to leave the vessel there for another year.