Government troops, backed by Russia and loyalist militia, launched a ferocious assault more than a month ago to retake the enclave on the outskirts of Damascus.
They have recaptured more than 90 percent of it and are draining the last rebel-held pockets through negotiated withdrawals brokered by Russia.
Two such deals have already seen thousands of rebels, their relatives and other civilians bused out of bombed-out Ghouta districts to Idlib, a northwestern province most of which still escapes government control.
The largest numbers have quit the towns of Arbin and Zamalka, and the adjacent district of Jobar, all controlled by the Faylaq al-Rahman Islamist faction.
The group reached a deal with Moscow on Friday and its implementation began the following morning with nearly 1,000 people boarding buses and leaving.
The numbers have grown steadily since, with the biggest convoy yet departing in the early hours of Tuesday with more than 6,700 people aboard.
They arrived Tuesday afternoon to the Qalaat al-Madiq area of central Hama province, a staging ground frequently used in such deals, AFP's correspondent there said.
That convoy brought the total number of evacuees from areas under Faylaq al-Rahman's control to 13,165 people.
The group's front man, Wael Alwan, has said as many as 30,000 people could be evacuated in all.
The evacuations were continuing on Tuesday, with more than a dozen buses entering the Faylaq-controlled pocket, said state news agency SANA.
Hereditary President-for-Life Bashir Pencilneckal-Assad
The Scourge of Hama...
has used such evacuation agreements to recover swathes of territory since the uprising against his rule began seven years ago this month.
They have usually begun with the military encirclement of an area, followed by bombardment and a ground operation before a deal is reached.