Officers also want to trace the owners and users of a Vauxhall Astra van which was parked opposite her house at about 9pm on the evening of March 18, and a light hatchback car, possibly a Ford Focus, which braked unexpectedly in her road at 5:42am the next morning.
Detective Superintendent Dai Malyn, who is leading a new review of the case, said:
"What we want to know if why did that vehicle brake, who was in it, and did it stop to give her a lift to work."
North Yorkshire Police launched a review of the investigation last year after a new major crime unit was established by the force.
Detectives investigating the disappearance of chef Claudia Lawrence are searching for two men seen near her home.
The first man was seen outside her house in Heworth Road by a witness at 6.45am on the morning of Miss Lawrence's disappearance - March 19, 2009 - and was described as 55 to 65, with grey mid-length hair, and wearing a three-quarter length sandy coloured mac.
Police also want to find a man in his 30s, about 5ft 8in tall, with very tidy brown hair with a fringe, who was seen in the road in the week before her disappearance.
"It looked as if he was looking for an address, he was on the phone, and he had a rucksack on his back," Detective Superintendent Dai Malyn said.
"He appeared to walk up to a green door, which we thought was Claudia's door, and a lady at the door appeared surprised, but then let the man in."
Miss Lawrence was 35 when she was reported missing by her father, Peter, on March 20 2009.
She was last seen at around 3.05pm on March 18, walking back towards her home, and that night she spoke to both her parents on the phone.
It is thought something happened to her after she left for work early on March 19.
New DNA evidence from items recovered in her home have been recovered
Claudia's GHD hair straighteners were missing from her home
The DNA profile of an unknown man on a cigarette butt in her Vauxhall Corsa had been found - the car was in the local garage when she disappeared, but the DNA has not matched any man who had yet come forward
Detective Superintendent Dai Malyn, who is leading the new review of the case, said it was reasonable to consider the DNA profile may be linked to the man spotted by a witness smoking with a woman on the morning Miss Lawrence disappeared. He said:
Despite numerous high-profile appeals, neither the man or the woman have been traced.
In particular, the man who smokes with his left hand could be significant to the investigation in relation to the male DNA profile found on a cigarette butt in Claudia's car.
Further details of the police investigation will be released on the BBC's Crimewatch programme tonight.
Detectives investigating the disappearance of Claudia Lawrence said a new forensic examination of her house has uncovered the fingerprints of people who have not come forward five years after the university chef disappeared.
Senior officers announced a series of new lines of inquiry on the fifth anniversary of the day Miss Lawrence failed to turn up for work at York University in 2009.
Detective Chief Superintendent Simon Mason from North Yorkshire Police denied reports that his team had identified a prime suspect as they issued a new appeal:
I need to make it absolutely clear there is no prime suspect in this case. Nor is there any individual who can be deemed to be classified as of a suspect status.
There will be "no resolution" for the family of missing Claudia Lawrence until they know why she vanished from her home five-years-ago.
Speaking at a press conference in York ahead of a fresh appeal for information on her whereabouts, Mr Lawrence said:
No-one knows, unless they're in the same position as us, what it means every day and, in fact, most nights - wondering, worrying, 'where are you Claudia, are you safe, is someone holding you, have you been hurt' - even, of course, 'are you alive'.
It's now five years, which is over 1,800 of those days and nights. It is an awfully long time for anyone to be in this position. And it eats into you.
It's actually like a cancer and it just gets worse and worse because there's no resolution until we know what happened to Claudia.
The new search of the house where Claudia Lawrence lived alone in Heworth Road, York, follows the formation of a new Major Crime Unit at the North Yorkshire force.
But senior detectives stressed the move has not been prompted by any new lead.
Detective Superintendant Dai Malyn, who heads the Major Crime Unit, said: "There's no new smoking gun or startling piece of evidence. I wouldn't want people to believe that's the case. This is just part of the review process."
Mr Malyn said the search, which could last a fortnight, was the beginning of a review as part of the new team's remit to look at important cold cases.
He said: "Most cold case review work considers forensic re-evaluation as techniques advance and this case is no different."