The group says it is starting a probe into the cause of the attack, and said it is ready to cooperate with an global investigation to determine who carried it out.
The blast hit buses in the Rashidin area on Aleppo's outskirts.
Mohammad Darwish, a doctor who has been evacuated from Madaya, witnessed the explosion.
Hours after the explosion, the transfer resumed - as dozens of buses, starting with the wounded, left to their respective destinations.
More than 64,000 regular people are "caught in a cycle of day by day brutality and hardship", it said.
The blast earlier Saturday killed at least 39 people and wounded dozens.
Microsoft confirms Windows' NSA hacking loopholes have been patched
And, if the response of the security community is any indication, America's most famous whistleblower has reason to be alarmed. Some of the targeted Windows versions, such as Windows Vista and Windows 2008, have fallen out of support.
"There's no drinking water or food", Madaya resident Ahmed, 24, told Reuters news agency earlier, speaking from the bus garage in Aleppo where he and others had been reportedly waiting since Friday night.
Rebels say Damascus ruptured the terms of the arrangement expedited by Iran and Qatar, blaming the legislature for attempting to bring out more follower warriors from than concurred. He accused the government or extremist rebel groups of orchestrating the attack to discredit the opposition.
Thousands of evacuees from both sides of Syria's civil war have been stuck in hostile territory since Friday.
Residents of al-Foua and Kefraya, the Shia villages, were waiting in the Rashidin area.
The agreement is one of several over recent months that has seen President Bashar al-Assad's government take back control of areas long besieged by his forces and their allies.
The much criticized deal is to transfer thousands from besieged pro-government and anti-government areas.
Trump says US-Russia relations may be at "all-time low"
President Vladimir Putin received Lavrov and Tillerson at the Kremlin Palace, spokesman Dmitry Peskov told reporters later. But such hopes appeared optimistic as the diplomats outlined their sharply diverging views on Syria.
Critics say the string of evacuations, which could see some 30,000 people moved across battle lines over the next 60 days, amounts to forced displacement along political and sectarian lines. The opposition Syrian Observatory for Human Rights put the death toll at 43, adding that it would likely rise because of the extensive damage.
A big auto bomb has impacted an escort of mentors conveying evacuees from government-held towns in Syria, allegedly murdering 16 individuals.
The state TV channel said the explosion was caused by a vehicle bomb.
Photos carried by regime media showed what appeared to be the aftermath of the explosion, with bodies lying on the ground and fires belching out thick black plumes of smoke. Only meters away, hundreds of evacuees from pro-rebels areas also loitered in a walled-off parking lot, guarded by government troops.
A Lebanese TV channel close to the Syrian government showed footage of new buses arriving at an evacuation point near the site of the explosion. It was not immediately clear if al-Hakim was the same French national USA officials said was killed in an airstrike in Syria in November 2016.
The British-based Syrian Observatory for Human Rights (SOHR) said the delay was because opposition groups from Zabadani, one of the towns included in the deal, had not yet been granted safe passage out.
Half girlfriend: arjun kapoor & shraddha kapoor lock lips
Shraddha looks lovely in the trailer and her acting looks better than her previous films. This is the first time Arjun and Shraddha are teamed up together for a film.
He said it is not clear what hinders the completion of the evacuation.