More than 210 have been pulled from the 16-storey Wei-kuan building - the only high-rise in the southern city of Tainan to crumble completely when the 6.4 magnitude quake struck before dawn on Saturday, killing more than 40 people.
But the city government and rescuers estimated more than 100 others could still be buried in the rubble.
Tainan mayor William Lai ordered rescuers to start using diggers and extractors to remove giant concrete slabs to better detect signs of life, which they have found in three different areas.
"It's approaching the 73rd hour and relatives are getting more anxious as time passes by and expect more. They hope the rescue team can make further moves," he said at the scene early Tuesday.
A tearful woman broke through the rescuers' lines and threw herself down as the mayor was walking nearby.
"I beg you to save us. Our family still have three people trapped inside," she told him.
Angry, distraught relatives waiting for news of their loved ones repeatedly interrupted the mayor as he gave a briefing on the progress of the rescue operation.
"Some of my family are in hospital and the bodies of others are in a funeral home. I am exhausted and I have to be here to wait for news of those still missing," one shouted.
"You keep asking us to wait and we have to get the latest information from the media. It's so confusing. We are going to break down," another man complained.
One woman said she was losing hope after three days of waiting for news of her loved ones.
"My brother and sister-in-law are trapped in Building A at the bottom of the wreckage. I feel like they've given up on them," Ms Cheng Ya-ling told AFP.
"I've been waiting since Saturday in freezing weather at night and I have blankets, how are they going to survive buried down there?" she said.
"I'm losing hopes and losing faith in the rescue. If there's no miracle of them coming out alive, I only hope they had passed away quickly and hadn't suffered long."
Use of heavy machinery had been repeatedly delayed after rescuers detected signs of life in upper parts of the toppled structure.
There have been some dramatic rescues, including a eight-year-old girl and three others pulled from the wreckage on Monday.
While the rescue operation was underway on Tuesday, the island was jolted by a 4.9-magnitude quake off the eastern city of Hualien city, but no damage or casualties were reported.
President Ma Ying-jeou said on Monday there was still hope of survivors, even beyond the 72-hour window.
"We will carry on until the last second. The golden 72 hours of rescue is the standard but there are many exceptions," said Mr Ma after visiting two survivors in hospital with bone fractures.
Cranes, drills, ladders, sniffer dogs and life detection equipment are being used to locate those trapped, but emergency workers and soldiers have also had to shore up the ruins to avoid further tragedies.
Prosecutors have launched an investigation into the collapse of the building as cans and foam were found in the rubble, sparking speculation construction had not followed the national building code.
Taipei mayor Ko Wen-je said the killer quake would speed up urban renewal projects in the capital which had been hampered for various reasons.
"It would cause huge risks for our citizens should any earthquake of the same scale hit the Taipei area," he told reporters.
The quake struck the weekend before Chinese New Year, when many relatives would have joined families to enjoy the holiday.
Taiwan lies near the junction of two tectonic plates and is regularly hit by earthquakes.
The island's worst tremor in recent decades was a 7.6 magnitude quake in September 1999 that killed around 2,400 people. - AFP
Did you find this article insightful?