Ellie Butler was killed by Ben Butler in the family's Sutton home in 2013, 11 months after she returned to his care.
Senior Coroner Selena Lynch said her "preliminary view" was there were issues that needed to be investigated.
But she stressed she would only make a final decision after receiving submissions and reading case papers.
Ellie's grandfather Neal Gray, who was in court, has called for a full public inquiry into her death.
Ben Butler, 36, was jailed for life with a minimum term of 23 years in June. His partner, and Ellie's mother, Jennie Gray was sentenced to 42 months for child cruelly and attempting to pervert the course of justice.
The hearing in Croydon has been considering whether an inquest into the six-year-old's death should be resumed.
Ms Lynch said it was "very unlikely" any new facts about the events of Ellie's death would be revealed but an inquest could look at "wider issues".
"The management of her [Ellie's] care by other agencies such as social and medical services...in my view do fall to be investigated", she said.
Sutton Council said it would "cooperate fully with the course of action the Coroner decides to take".
It said: "An independent Serious Case Review was carried out by Sutton's Safeguarding Children Board and it found we did everything we could in the circumstances."
Neal Gray, who looked after Ellie before she was returned to her parents, has called for an investigation into why family court judge Dame Mary Hogg "exonerated" Butler of causing head injuries to her when she was a baby.
He said he remained "deeply concerned" about decisions made by a family court that had put Ellie in a "dangerous situation" and prevented agencies from acting on "clear warning signs".
"My aim is to get answers and to make sure that lessons are learned so that this cannot happen to another child," Mr Gray said.
Ms Lynch said a hearing could not review previous judicial decisions, judges' "thought processes" or findings.
But she said the "framework" of the way decisions were reached could be examined, as well as the way courts interact with other agencies.
Butler has lodged an application to appeal against his conviction and sentence.
The way social services handled the case of a six-year-old girl who was murdered by her father should be investigated, a coroner has said.