Sturgess died on Saturday and her partner Charlie remains in hospital but is said to be "conscious".
He had moved from critical to a "serious, but stable" condition, said Lorna Wilkinson, director of nursing at Salisbury hospital.
She said that he still "has some way to go", but the staff is optimistic about his recovery.
Detectives are yet to publicly identify any suspects in connection with the nerve agent attack on former Russian spy Sergei and Yulia Skripal in Salisbury, Wilts, on March 3.More news: Dan Martin powers to Tour stage victory
Describing the difficulties of the investigation, Metropolitan Police Assistant Commissioner Neil Basu said: 'There are multiple scenes, there are only limited numbers of scientists and forensic medical officers - all of whom are volunteers by the way - to do what is a really, really unsafe piece of work for obvious reasons.
"Any contact officers have with Charlie will be done in close consultation with the hospital and his doctors".
Neil Basu, Britain's most senior counter-terrorism officer said it might never be possible to definitively establish a link between the death of Sturgess and the poisoning of the Skripals.
On Sunday, police issued a statement saying there had been no evidence that either Rowley or Sturgess had been to any decontaminated sites from the nerve agent attack on Skripal and his daughter.
"We are cross about the scathing comments".More news: Deadly Mexico quake unearths ancient temple
Police now believe the two handled a contaminated item with a high dose of Novichok, although they have not been able to confirm if it was from the same batch as the one used in the attempted murder of the Skripals.
"This is a very rare substance banned by the global community and for there to be two separate, distinct incidents in one small English county is implausible to say the least", Basu added.
Novichok is the name for a group of Russian chemical weapons designed by the Soviet Union in the 1990s, and are among the most toxic substances ever created. I'm praying for Charlie to come back, "responded Ben Jordan, 27, who was housed in Salisbury's homeless shelter John Baker House where Dawn Sturgess, who has since been evacuated by police, lived".
"I simply can not offer any guarantees", said Assistant Commissioner Neil Basu, who is leading the investigation.
"Dawn's death only strengthens our resolve to find out exactly what happened and who was behind it", Home Secretary Sajid Javid, the interior minister, told parliament after chairing an emergency cabinet meeting on the incident.More news: Now-Hurricane Chris Poses Possible Threat to Nova Scotia and Newfoundland-Labrador
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