Tens of thousands of civilians have streamed out of the enclave outside Damascus in less than a week.
Russia-backed regime forces have retaken most of the former opposition bastion since February 18, slicing remaining rebel-held territory into three separate pockets.
The displaced have gathered in regime-controlled territory outside the enclave, including in Adra to its north.
In one makeshift shelter, AFP journalists saw hundreds of people assembled on thin bedding under a tarpaulin sheet, with donated blankets piled beside them.
Dozens of people -- including women and children -- queued outside limited bathroom facilities.
"People may have escaped fighting, fear and insecurity but they find themselves in a place without anywhere to wash themselves. This should not be," Zaatari added.
Since March 15, more than 70,000 people have fled the enclave near Damascus after weeks of bombardment, says the Britain-based Syrian Observatory for Human Rights.
The UN says shelters have received over 50,000 in the past week.
"This crisis must be managed in a different way and the solution is to speed up people's return home," Zaatari said.
"The solution is to empty these shelters of inhabitants, as soon as possible, and to keep residents in Eastern Ghouta," Zaatari said, if security conditions allow.
"Keeping people in their homes and aid reaching them there is easier than bringing them to these public places," he said.
Before the regime assault, some 400,000 civilians in Eastern Ghouta had lived under government siege since 2013, facing severe food and medicine shortages.
Zaatari was also critical of the humanitarian situation for tens of thousands who have fled a Turkey-led advance on the northern region of Afrin.
Pro-Ankara forces swept into the Kurdish region's main city -- also named Afrin -- on Sunday.
The United Nations says around 100,000 people have fled the region since the start of the assault on the Kurdish People's Protection Units (YPG) militia there on January 20.
"We cannot access Afrin as it's an occupied region," Zaatari said, adding attempts were being made daily to try to reach people.
He said areas hosting the newly displaced outside Afrin were coming under "increasing pressure".
More than 350,000 people have been killed since Syria's war started in 2011 with the brutal repression of anti-government protests.