Some 18 army officers, including a major-general, have already been charged in connection with the case.
The defence minister has also resigned.
Hung Chung-chiu, 24, was only three days away from completing his military service when he died of organ failure brought on by heatstroke.
Corporal Hung had been held in solitary confinement for bringing a mobile phone with built-in camera onto his military base – and was then subjected to arduous punishment exercises in the hot sun.
His death has generated outrage across Taiwan and damaged the standing of the army, which is already struggling to find enough volunteers as it tries to phase out conscription, says the BBC’s Charles Scanlon.
President Ma Ying-Jeou has publicly apologised and promised justice for the family.
But protesters in the streets of Taipei, many dressed in white to symbolise truth, said they were not convinced by the sincerity of the investigation in the case.
Other families have also come forward to demand justice for their sons who died during military service.
“I am mourning for Hung Chung-chiu and I want the truth. I hope there won’t be any more abuse and death like his in the military,” protester Jenny Tan told the AFP news agency.
It is a crisis of confidence for Taiwan’s army of more than 200,000, as it tries to maintain the credibility of its defensive stance against the threat from China, our analyst says.
All Taiwanese men aged between 18 and 36 are required to complete one year’s military service, although there are plans to move to an all-volunteer force.