A convoy of around 20 buses crossed the front line in the early morning headed for rebel-held territory elsewhere in northern Syria, after around 350 people got out during the night.
They were the first departures since Friday when the government suspended evacuations insisting that people also be allowed to leave two northwestern villages under rebel siege.
A medic said the latest evacuees were in a "terrible state" after their departure was delayed for hours in temperatures well below freezing, compounding their plight from months of siege and bombardment by the army.
The new evacuations from the devastated enclave came as a monitoring group said rebels had finally allowed convoys out of two villages they have besieged since spring last year.
Around 500 people left in a dawn convoy out of Fuaa and Kafraya, the Syrian Observatory for Human Rights, a Britain-based monitoring group, said.
Government demands for evacuations from the two Shiite-majority villages had put the whole process on hold for days, and on Sunday rebels attacked buses sent to bring people out, killing one of the drivers.
Around 350 people in five buses made it out of Aleppo during the night after Russia and Turkey urged the Syrian regime to allow a convoy of buses to pass its final control point, the Observatory said.
It was regime ally Moscow and rebel supporter Ankara that brokered a first evacuation deal last week to end a blistering month-long government assault on the one-time rebel bastion in east Aleppo.
"Five buses carrying the evacuees arrived from besieged parts of east Aleppo," said Ahmad al-Dbis, who heads a team of doctors and volunteers coordinating evacuations.
"They were in a terrible state," Dbis told AFP.
"They hadn't eaten, they had nothing to drink, the children had caught colds, they were not even able to go to the toilet."
Dbis said he saw families wrapped in several layers of coats getting off the buses.
One young boy was biting into an apple while aid workers distributed packs of bottled water to his family.
Evacuations were suspended on Friday, a day after convoys of people had begun leaving the rebel sector under a deal allowing the regime to take full control of the battleground city.
The main obstacle to a resumption had been the dispute over how many people would be evacuated in parallel from the two Shiite villages.
A rebel representative said that hundreds of people would also be evacuated from Zabadani and Madaya, two rebel towns near the Lebanese border under siege by the army, as part of the deal.
Iran's official news agency IRNA said the foreign ministers of Russia, Turkey and Iran would meet in Moscow on Tuesday to discuss the situation.
Moscow, which has carried out an air war in support of the Damascus regime since September last year, had threatened to veto a draft of the resolution calling for monitors to oversee the protection of civilians.
But after four hours of closed-door consultations on Sunday it gave the French-drafted text its guarded support.
US ambassador Samantha Power anticipated member states would vote "unanimously" at 1400 GMT.
Families have been sheltering at night in freezing temperatures in bombed out apartment blocks in Aleppo's Al-Amiriyah district, the departure point for evacuations.
An AFP reporter visited a hospital in the rebel sector where patients lay on floors without food or water and almost no heating.
Aleppo has seen some of the worst violence of the nearly six-year war that has killed more than 310,000 people.
A physiotherapist, Mahmud Zaazaa, said only "three doctors, a pharmacist and three nurses" remained in the area.
An official said more than half of Aleppo's buildings had been destroyed or seriously damaged since the rebels overran the eastern sector in summer 2012.
UN envoy Staffan de Mistura estimated that as of Thursday around 40,000 civilians and perhaps as many as 5,000 opposition fighters remained in Aleppo's rebel enclave.
Before evacuations were suspended around 8,500 people, including some 3,000 fighters, left for rebel-held territory elsewhere in the north, the Observatory said.