The Indian-origin nurse, who apparently took her life after being duped by a prank call from two Australian radio hosts, had left a note for her family members.
But her family members still believe there are unexplained circumstances behind her death, Labour Member of Parliament Keith Vaz has said.
"What the family needs are the full facts. There are unexplained circumstances. The family wants to know everything," Vaz said, without revealing the content of the note.
"The hospital needs to be more proactive, a full inquiry is needed and the family needs to be included in that," he added.
Saldanha, 46, died after a suspected suicide last Friday after she was fooled by the radio hosts, who made a hoax call to the King Edward VII Hospital in Marylebone, central London.
She was the senior nurse on duty when the phone call was made.
Although the Marylebone hospital authorities have said they will conduct an internal probe into the matter, the family says this is not enough, and demands answers from them over the circumstances that led to her death.
Vaz, who is representing the family, on Tuesday met the hospital chairman Lord Glenarthur to demand a full probe into her death.
"What the family needs are full facts. What the chairman of the hospital said to me was that there are inquiries going on in the hospital. That is not sufficient for the family.
There are unexplained circumstances. The family wants to know everything. All the facts, fully and clearly," Vaz said.
"They were in the dark about the hoax call. They hadn't watched the news or seen anything about it," he said, seeking support for the victim's family.
Bosses at the Australian radio station 2Day FM, whose presenters made the call, were under pressure to come forward and explain why they gave the go-ahead for the prank.
Southern Cross Austereo, the company which owns the station, said it will donate 3,20,000 pounds to a memorial fund set up in Saldanha's name by King Edward VII Hospital, and added it has cancelled this year's Christmas party.
"We hope that by contributing to a memorial fund we can help provide the Saldanha family with the support they need at this very difficult time," said Rhys Holleran, Southern Cross Austereo's chief executive.
Meanwhile, a local newspaper quoting a family source has claimed that the victim's husband Benedict Barboz had tried to track her down in the hours before her suspected suicide.
Barboza had not heard from his wife for two days, so he called up one of her friends to check on her, the Mirror reported.
Barboza called at Jacintha's Central London flat at around 9.30 pm (local time) on Thursday, a day before she was found dead, but he did not hear anything back from the friend.
Jacintha would normally be in regular contact with her husband at the family home in Bristol, but she stopped phoning or taking calls after news of the prank call broke, the paper quoting the family source said.
"Ben was so worried he called a friend who then knocked on the door. But he never heard back from that person. Then he finds out that his wife's been found dead the next day," the source said.
The Mirror revealed on Tuesday that Benedict was angry with the hospital for the way it handled the aftermath of the hoax.
The family said it learnt there were no switchboard staffers at the hospital at certain times of the day, leading to the phone system automatically diverting calls to a senior nurse.
The hospital claims that it supported her after the hoax and did not discipline her in any way.
