"The evacuation continued throughout the night and it will continue all through today and for so long as there are people who want to leave," International Committee of the Red Cross spokeswoman Ingy Sedky said.
She said ambulances and buses had been travelling back and forth all night from the last pocket of rebel territory in east Aleppo to areas held by the rebels elsewhere in the province, with vehicles unloading passengers and immediately returning to pick up more.
Initially the vehicles had all moved together in a single convoy, but overnight each vehicle returned to collect more evacuees as soon as it was empty, she said.
"That means it is difficult for us to know exactly how many people have left so far, but there will be an assessment by the end of the operation," she said.
Apart from the green government buses and ambulances transporting evacuees, some residents left in their own vehicles, said Ahmed al-Dbis, the head of a unit of doctors and volunteers coordinating the evacuation of wounded people.
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"There are people coming in their private vehicles," he told AFP from an evacuee arrival point in the west of Aleppo province.
"Many families are coming with their household belongings in pick-up trucks.
"These people are gathering here and then going further west in the province and from there either to camps or to relatives or friends living in the area."
He estimated around 6,000 people had arrived from east Aleppo so far, among them at least 250 wounded people. Another 250 or so needed medical attention for illnesses.
Some of those arriving required emergency care and were transferred to Turkey, he added.
East Aleppo, in rebel hands since 2012, was surrounded by government troops in mid-July.
The blockade caused acute shortages of food and medicines, and a devastating air and artillery bombardment destroyed most hospitals and clinics, leaving residents with little access to medical care.
The army seized more than 90 percent of east Aleppo during an assault that began in mid-November, and is expected to declare its full control of the city as soon as the evacuation is complete.