The military have been providing support since Friday and Rudd said the police would continue to be given all the resources they needed to conduct the investigation.
Yulia Skripal, 33, is critically ill in hospital along with her father Sergei after they were poisoned with a deadly nerve agent last Sunday.
Sergeant Nick Bailey, one of first officers to find the pair slumped on a bench outside a shopping centre and who was later hospitalised in intensive care, was sitting up and talking on Thursday, according to police.
Sergei Lavrov expressed resentment at suggestions that Moscow was behind the nerve agent attack on the Skripals, calling such claims "propaganda".
Sergei and Yulia Skripal are both in a serious condition in hospital, after being exposed to a nerve agent, and becoming unwell in a park in Salisbury last week.
The military was set to "remove a number of vehicles and objects from the scene in Salisbury town center", police said, adding the public should not be alarmed.
The operation on the street where Mr Skripal lived has ramped up considerably.
Rudd visited several of the sites cordoned off by investigators before heading to Salisbury District Hospital.
Earlier Friday, members of the armed forces removed a police vehicle used in the initial response from Salisbury district hospital.
The authorities did not provide details, saying only that they had not exhumed any bodies, but the forensic activities at the London Road Cemetery intensified speculation about the poisonings.
Earlier this week, a spokesman for the Russian embassy in London described comments by UK Foreign Secretary Boris Johnson - who vowed a "robust response" in the event that state involvement was proved - as "strongly anti-Russian". They called on anyone who visited the area on Sunday, including a branch of the Zizzi restaurant chain or the nearby Bishop's Mill pub, to come forward with any information that might help them piece together what happened.
Officials are still investigating how the nerve agent was administered and lawmakers are probing a possible Kremlin link to the incident.
British media reported that he died while in St Petersburg, Russia.
Skripal was a former colonel in Russian military intelligence who was jailed in his country for betraying agents to Britain's MI6 secret service. In 2010, President Dmitry Medvedev pardoned him, after which, in the framework of exchange of spies with the USA, he was sent to the United Kingdom, where he was granted asylum. His condition doctors assessed as "serious but stable".
The attack has been likened in Britain to the assassination of ex-KGB agent Alexander Litvinenko, a critic of Russian President Vladimir Putin, who died in London in 2006 after drinking green tea laced with radioactive polonium-210.
He told Sky News: "Regardless of whether the Russians are involved in this case, they are fighting a new Cold War against us and you need to wisen up to that fact".
A British public inquiry said Litvinenko's murder had probably been approved by Putin and carried out by two Russians, Dmitry Kovtun and Andrei Lugovoy. The conclusion was angrily dismissed by the Kremlin as a politically motivated smear.
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