Most of the new arrivals are from Yei, a city that lies close to the borders of Uganda and the Democratic Republic of Congo. They say they received letters warning them to evacuate the town in anticipation of conflict between rebels and government forces.
Others are from Eastern Equatorial, with smaller numbers from the capital Juba and Upper Nile state.
The refugees have also reported that armed groups operating across different parts of South Sudan are looting villages, murdering civilians and forcibly recruiting young men and boys into their ranks. They add that armed groups operating on roads to Uganda are preventing people from fleeing South Sudan.
Charlie Yaxley, the spokesperson of the United Nations High Commission for Refugees - UNHCR in Kampala says that many of the children registered at the reception centres have lost one, or both of their parents.
"Many took advantage of the opportunity to flee alongside Ugandan military convoys evacuating Ugandan nationals," Yaxley said in a statement issued this afternoon.
Kenya has also reported the arrival of 1,000 refugees in the same period, while 7,000 have fled to Khartoum. In total, 60,000 people have fled the country in less than one month, bringing the overall number of South Sudanese refugees in neighboring countries since December 2013 to nearly 900,000.
Both Kenya and Uganda are reporting rising cases of severe malnutrition, particularly among very young children. Those found to be suffering are being placed on food nourishment programmes to bring them back to health.
Yaxley adds the generalized collapse in the protection of civilians is also affecting many of the 250,000 refugees, mainly from Sudan, Ethiopia, and the DRC. Aid agencies are unable to provide urgent help to needy populations.
With over 2.6 million of its citizens forcibly displaced, the world's youngest nation currently ranks among the countries with the highest levels of conflict-induced population displacement globally. Half the population relies on humanitarian aid.