Violence has erupted in the flashpoint town of Bor, capital of eastern Jonglei state, and in Torit, capital of Eastern Equatoria.
President Salva Kiir has accused ex-vice-president Riek Machar of staging the coup - a claim he denies.
The UN called for political dialogue to end a crisis that has left hundreds dead and sparked fears of a civil war.
On Wednesday, Mr Kiir said he was willing to enter into talks with Mr Machar but that he did not know what the result would be.
He said: "This is a political crisis and urgently needs to be dealt with through political dialogue. There is a risk of this violence spreading to other states... [and] we have already seen some signs of this."
Mr Ban confirmed that some 20,000 people had taken refuge in the UN mission in Juba.
The mayor of Bor, Nicholas Nhial Majak, told the BBC's Focus on Africa programme that violence had spread there from Juba, where it began on Sunday. Bor is about 150km (90 miles) north of the capital.
The mayor said troops loyal to breakaway army commander Peter Gadet had raided the town and there was "a lot of fighting".
"There is a lot of panic in town - and we really want the authorities to send more troops to quickly restore hope for the people," he said.
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