Yei is situated some 150 kilometres southwest of Juba, near the borders with Uganda and the Democratic Republic of Congo, and until recently had been spared much of the violence plaguing the world's youngest nation.
But the security situation there has deteriorated rapidly since July.
The UN refugee agency said now around 100,000 people - many of whom had fled into the town in search of safety - could no longer leave after government troops surrounded the area.
He pointed out that more than 30,000 people had fled into Yei from surrounding areas following deadly attacks on civilians and looting of private property earlier this month.
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They joined several thousand displaced people who had arrived since mid-July, and as many as 60,000 town residents, he said.
UNHCR and other UN agencies had visited the town on September 27, discovering that tens of thousands of displaced people had taken refuge in abandoned houses and church compounds, Spindler said.
In Yei, the displaced "are facing a serious shortage of food and medicine," Spindler said.
He said humanitarian workers were hoping to soon bring desperately needed aid into the town, but acknowledged that the timeframe for such a delivery remained unclear.
South Sudan, which gained independence in July 2011, descended into war just two and a half years later when President Salva Kiir in December 2013 accused his former deputy Riek Machar of plotting a coup.
Since the fresh violence in July, more than 200,000 people have fled South Sudan, sending the number of refugees from the war-scarred nation past the one-million mark, UNHCR said.
Another 1.61 million people are displaced inside the country, it said.