Two toddlers found dead along with their parents did not suffer knife wounds, tests have shown.
Andrew and Vicky Case and daughters Phoebe, two, and Nereya, one, were found in Fordingbridge on Monday.
Tests on the girls' bodies have proved inconclusive but there were no signs they had been stabbed, police said.
It has emerged Mr Case had filed an alternative to bankruptcy, known as an Individual Voluntary Arrangement (IVA), at Salisbury County Court in 2006.
The arrangement, which usually lasts a period of about five years, is still listed as "current".
An IVA requires a person to pay back as much of their debt as they can afford, after which the balance is written off.
A spokesman for Hampshire Police said they were continuing to investigate the background and circumstances that led to the four deaths.
Tests carried out on Monday revealed that Mr Case, 33, a delivery driver, died as a result of hanging, while his wife died from knife wounds.
Police are investigating whether Mr Case killed his wife and children before hanging himself.
Post-mortem examinations on the children were carried out at the Royal Hampshire County Hospital in Winchester on Tuesday afternoon but further tests will be required.
The ambulance service had said on Monday that the girls were found stabbed.
Remembrance services are to be held at St Aldhelm's Church in Sandleheath and St Mary's Church in Fordingbridge on Monday.
A police cordon remains in place at the family home in Provost Street where the bodies were found, while forensic investigations continue.
The discovery of all four bodies was made by the children's grandmother, Linda Haskell, who raised the alarm.
The family of four were popular and well-known members of their community, residents have said.
Mrs Case ran St Aldhelm's children's nursery in nearby Sandleheath.
Neighbour Alayna Brooks said the family had just returned from a week's holiday in Weymouth, Dorset.
She described the little girls as "gorgeous" and the couple as "loved up".
Detectives have said they are not looking for anyone else in connection with the investigation.
Scenes of crime officers took away a Land Rover Freelander and a Volkswagen Golf on Monday evening.
The house where the bodies were found is set back from a row of cottages, next to a stream on the edge of the New Forest.
A small set of floral tributes have been laid at the scene.