Rebecca Williams, known as Bex, was rescued from the New Year's Day fire in which her boyfriend Cameron Logan died.
His parents were treated for smoke inhalation after the fire at their home in Milngavie, East Dunbartonshire.
Police said they still wanted to trace dog walkers and were seeking information about a dark-coloured car.
They want to speak to a man seen in the Craigton Wood area around the time of the fire walking a brown "pitbull" type dog, as well as another man walking along Craigton Road onto the West Highland Way with two Springer Spaniels.
Detectives have said these people could be potential witnesses.
The dark-coloured car was spotted in a layby on nearby Craigton Road near Clober Golf Club that had a door open and engine running around the time of the fire.
Ms Williams, a 24-year-old journalist at Global Radio, is being treated in hospital in Glasgow.
On Friday, her father Phillip Williams posted on Facebook that his daughter was "fighting hard".
Her condition later went from critical to "serious but stable" and NHS Greater Glasgow and Clyde confirmed on Sunday that Ms Williams was continuing to improve.
A woman who was seriously injured in deliberate fire which killed her boyfriend has spoken to detectives from her hospital bed.