Peter Stanley, 56, who lives in a flat in the mansion to the right of Mr Jefferies in Clifton, Bristol, said he, the landlord and Miss Yeates's live-in boyfriend, Greg Reardon, worked as a team to get the car on the road on the night of Friday, 17 December.
Mr Jefferies, 65, the former head of English at Clifton College, is said to have been told by Mr Reardon, 27, that he was going to Sheffield to visit his parents.
Police were last night granted a second extension to question him, and could now hold Mr Jefferies until 7pm on Monday.
Recounting the episode, Mr Stanley, 56, said yesterday: "It was a non-event at the time, but absolutely poignant now - what if we didn't get the car to start?"
Mr Stanley, joint co-ordinator of the community's Neighbourhood Watch scheme - with Mr Jefferies - said: "I got a phone call from Mr Jefferies. I was asleep at the time and it was early evening. He asked if his tenant could borrow my jump leads. I think Chris Jefferies wanted to borrow my car, but I said I didn't want to take all the power out of it.
"So he said, 'fine', and he then brought his car - I think it was the Chrysler."
Yesterday, Avon and Somerset Police said Mr Stanley was helping detectives with inquiries and that he was being treated as a potential witness and had not been arrested.
Miss Yeates, who disappeared a few hours after her boyfriend drove away on 17 December, was found strangled on Christmas Day near a quarry three miles from the basement flat she and Mr Reardon rented from Mr Jefferies, whose flat is two floors above their own.
The landscape architect had returned home after a night out in a pub with colleagues. She disappeared some time after returning to her flat in Clifton, the city's most affluent suburb.
Mr Jefferies, who bought the flat Miss Yeates lived in from a former colleague, who was jailed in 2008 for child abuse, was arrested at 7am on Thursday and police were yesterday granted a 12-hour extension to question him. On Wednesday he had given police what appeared to be a key piece of evidence about seeing Miss Yeates leaving her flat with two other people on the night she vanished.
Neighbours have described Mr Jefferies, who sports a distinctive mane of straggly white hair which he used to dye blue, as a "nutty professor type".
He was active in the community, playing a prominent role in the Neighbourhood Watch group, an enthusiastic activist for the Liberal Democrats in Clifton, and also a member of the Clifton and Hotwells Improvement Society campaigning to conserve buildings in the area.
But some former tenants and pupils have said Mr Jefferies had an obsession with Christina Rossetti, a 19th-century poet who often wrote about death, had a tendency to create unease by reportedly entering flats unannounced and for having made inappropriate sexually suggestive remarks to pupils.
One former tenant, who along with his wife rented one of Mr Jefferies' three flats in the four-storey Georgian building, was reported as saying: "He always seemed to be hanging about. If we left the flat he was always outside. On several occasions he even entered our flat unannounced. He acted surprised and left when my wife confronted him… I had a word with him and told him to only come in if it was prearranged."
Former pupils at 30,000-a-year Clifton College, whose alumni include John Cleese and the late actor Sir Michael Redgrave, said he was known as "The Strange Mr Jefferies".
One said: "He was fascinated by making lewd sexual remarks. It was disturbing."
Another added: "He used to touch people's hands and he'd say, 'Oh, you're very sweaty. That means you're sexually active'."