Clashes erupted in Bustan al-Qasr in the southwest and in nearby Izaa as the districts were shelled, the residents said, while also reporting fighting further south in Sukari, where many insurgents were holed up.
Overnight, two civilians were killed by shelling in the rebel-held Sakhur neighbourhood in the northeast, while nearby Hanano was also bombed, the Syrian Observatory for Human Rights said.
Syrian forces said they had secured the flashpoint Midan district on Monday after a week of fighting, although an AFP correspondent said some parts were still unsafe for residents to return.
The correspondent reported seeing nine bodies near the fourth zone of Midan district on Monday, which the army had blocked off to residents because snipers were said to be in the area.
Pro-regime Syrian newspaper Al-Watan said the army had "cleansed" the Midan district, which "opens the door for cleansing" nearby neighbourhoods, including Bustan al-Basha, Suleiman al-Halabi and Sakhur.
But the battle for Aleppo is very fluid, with both sides claiming gains in what is a guerilla war, Observatory director Rami Abdel Rahman said.
"When the army says they control an area, it is only temporary," Abdel Rahman told AFP. "They take districts and there are clashes with the guerrillas again, this is not real progress."
According to him, the army did not retake Midan, for the simple reason that the district was never in rebel hands in the first place. "The insurgents held only a police station and two or three streets."
But this applies to the rebels as well.
"They claim to control a checkpoint or a post, but then the army comes and destroys everything. That is not real control."
"The real superiority of the regime is its air force," he said.
Outside Aleppo, two civilians were killed in shelling on the town of Al-Bab and the town of Safira was also reportedly bombed, according to the Britain-based watchdog.
The violence followed a bloody day in which 137 people -- 72 civilians, 24 rebels and 41 soldiers -- were killed nationwide on Monday, according to the Observatory.