Separatists' forces late on Monday advanced on the presidential palace and captured two military camps near Aden global airport, security sources told AFP news agency.
In downtown Aden, most people stayed off the streets as government forces guarded the main road to Al Maasheeq palace where the government is based.
The fighting began on Sunday after a deadline set last week by the STC for Hadi to dismiss bin Daghr's government, which the STC accused of corruption and mismanagement.
It is the second such call in a week by the coalition, which warned of necessary measures to restore security and stability in Aden, yet without specifying concrete steps or if military action would be taken in the temporary capital of Yemen.
Separatist forces took over much of the city and by Tuesday evening they had surrounded the presidential palace, where the country's prime minister, Ahmed bin Dagher, and several other senior government ministers were holed up.
He said the STC forces rapidly advanced in the street fighting and imposed a tight besiege around the Presidential Palace in Aden's district Crater, where the government officials reside.
Soldiers stand guard as workers unload aid packages from a Saudi air force cargo plane, at an airfield Yemen's province of Marib.
"We call on all parties to refrain from escalation and further bloodshed".
Saudi Arabia's official press agency, quoting a coalition statement, said that the coalition had been "watching with regret all over the past two days that all parties have not responded to the calls for calm" and that it had requested "all parties to speed up the cessation of all clashes immediately".
The lack of any progress on political negotiations, Mr. Dujarric said, "only piles onto the misery of the Yemeni people". "Additional divisions and violence within Yemen will only increase their suffering".
Although Hadi remains in exile in Saudi Arabia, his administration and local allies nominally control about four-fifths of Yemen's territory.
Hashem Ahelbarra, an Al Jazeera correspondent who has reported extensively from the country, said the latest developments were a "game changer" for the coalition, "but for more than one reason". They want to bring back the independent country of South Yemen.
If this happens you will have two Yemens.
"The Houthis are consolidating their grip in the north and so are the secessionists in the south".
President Hadi issued a statement ordering the military to secure the city and assuring the population that his government was capable of handling the situation.