Local prosecutors have launched an investigation into possible manslaughter over Sunday evening's accident near the town of Avellino on the busy highway between Naples and Bari in southern Italy.
The ANSA news agency said the manslaughter probe would look into the possible role of the driver, as well as the state of the coach and the crash barrier on the highway.
ANSA said the driver's body would be examined for the possible presence of alcohol or drugs while traffic police have seized the vehicle documents from the coach operator Mondotravel.
Relatives and friends gathered at a nearby school where the bodies of their loved ones were laid out in the gymnasium, ahead of funerals to be held on Tuesday.
Rescuers have cleared the last of the wreckage from a wooded area off the road, where a row of beige seats patterned with blue swirls had lain, streaked with blood.
The coach, carrying 48 people including children, rammed several cars before it plunged off a viaduct through a crash barrier and down a slope about 50 kilometres from Naples in an area described as an accident black spot.
President Giorgio Napolitano described the accident as "an unacceptable tragedy" and called for improved road safety standards.
Police said 38 people had died, including the driver, although Transport Minister Maurizio Lupi had put the number of dead at 39.
Another 10 passengers were injured, along with another nine people in cars hit by the coach before it careered off the road.
The group was returning from a pilgrimage to Pietrelcina in the Campania region of southern Italy, the birthplace of Padre Pio, an Italian priest canonised in 2002 and worshipped in the country's south.
A small wooden cross was left by a wellwisher at the site of the crash, propped up near a bunch of roses.
Passengers' belongings had littered the ground, including a hat, shoes, and a child's teddy bear.
The accident was the deadliest in western Europe in a decade and the worst in Europe since an October 2010 incident in Ukraine when 45 people died.
Prime Minister Enrico Letta, on a visit to Athens, said it was a "very sad time" for Italy and observed a minute's silence in honour of the victims before addressing a conference in the Greek capital.
"This tragedy has profoundly moved our country... it is an open wound," he said. "I am grieving for and express my profound sorrow to the families of the victims".
The prime minister is due to visit the crash site on Tuesday, after returning from Greece.
Rescue workers pulled 33 bodies from the wreckage and found three more thrown from the vehicle as it plunged 30 metres down a slope.
Another two died in hospital of their injuries.
Photographers at the scene said about a dozen wrecked cars were strewn across the motorway, which had been closed to traffic.
"Looking down from the overpass, the scene of the tragedy: some 30 bodies covered by white sheets, lined up along the roadside," said Cesare Abbate of ANSA.
One survivor, quoted by his uncle who met him in hospital, reported hearing a tyre exploding and that the driver had been unable to control the vehicle.
The last major coach accident in Europe was in March 2012 in Switzerland, when a coach carrying Belgian schoolchildren home from a skiing holiday crashed, killing 28 people, including 22 children.
Funerals for the Italians will take place in the town of Pozzuoli where many of the victims lived.
A party to celebrate the unveiling of the Naples football squad wascancelled out of respect for the victims, and a minute's silence was to be observed before the team's friendly against Turkish team Galatasaray in Sao Paulo, Brazil.
The accident came just days after a train crash in Spain last Wednesday which killed 79 people, the deadliest rail disaster in the country in decades.