Smith's family and the presiding coroner want a broader focus that includes how Smith was treated in prisons outside Ontario.
The federal government and a handful of doctors want a narrowly focused hearing, arguing the inquest has no jurisdiction outside the province.
The arguments should have begun on Tuesday, but the government held up proceedings.
It spent the day pressing the coroner to ban publication of prison videos of Smith, then said it would go to court when the coroner refused the ban or to delay proceedings further.
The 19-year-old Smith choked to death in her cell in Kitchener, Ont., five years ago.
The video issue came to the fore because a handful of doctors — backed by Correctional Service — are challenging the scope of the inquest.
They argue Dr. John Carlisle's authority ends at the Ontario border.
Smith's family — backed by the province's child advocate, prisoner advocates and prison guards — argues the inquest must examine her treatment in institutions outside Ontario.
Smith spent her final year in solitary confinement, shunted 17 times among nine different prisons in five provinces with little treatment for her mental illness.
The family plans to use the videos and other materials to argue the inquest should be broad.
The videos show Smith being physically restrained for hours at a time, at one point strapped to a gurney in a wet security gown.
They also show staff at the Joliette Institution in Montreal giving her intravenous drugs without her consent. On one occasion, guards in riot gear surround the handcuffed Smith as she is injected.
At another point, they use duct tape to restrain her.