A senior detective has said he is talking to police forces across the UK about unsolved crimes and missing people because he believes a taxi driver convicted of murdering two young women may have killed others.

Chris Halliwell, 52, faces a full life term after being convicted of the murder or Becky Godden, a sex worker who vanished, aged 20, in 2003 from Swindon. He is already serving life for the murder of Sian O’Callaghan, 22, an office administrator whom he abducted from a nightclub in the Wiltshire town in 2011.

DS Sean Memory, the senior investigating officer in the Godden case, said Halliwell may have killed others. “I’m definitely concerned. We know that Becky died in 2003 and Sian in 2011. What I don’t understand is why there is that gap and how he can turn from a mild-mannered taxi driver taking young vulnerable women home and on other occasions turn into a killer,” he said.

Memory told BBC Radio 4’s Today programme he did not have any lines of inquiry. “I do know that Halliwell was very forensically aware. When he killed Sian he removed items of her clothing, he tried to get rid of fibres and he tried to get rid of items from his car. I definitely can’t rule out he committed other offences.

“I appeal to him to say: ‘You’ve put two families through absolute hell. This is the time to come forward and if there are any other offences speak to me now.’”

Memory referred to an odd exchange he had with Halliwell when he interviewed him in February last year. “He tried to strike a deal with me that if he would clear up Becky’s murder that I would never interview him about anything ever again.

“Of course I couldn’t give him that guarantee in case there are other rapes or murders. He said I didn’t go far enough in the guarantees I’d offer him. It’s a strange deal to try to strike if there are no other victims. He wanted absolute immunity from prosecution for the rest of his life.”

Memory added: “I will now go away and look at the timeline for Christopher Halliwell. He’s been a taxi driver up and down the country, he’s been a ground worker up and down the country. I’ll look to other police forces to find out if they have missing people in similar circumstances and see if any of that fits into any of the evidence I have.”

In the mid-80s, while he was in prison for an unrelated offence, Halliwell asked an inmate how many women a person needed to kill before being considered a serial killer.

And when police searched a remote pond where Halliwell dumped O’Callaghan’s boots after stabbing and strangling her in 2011, they discovered dozens of scraps of material that may be clothing.

In common with many violent sexual offenders, Halliwell had a fascination with hardcore pornography, including child abuse and bestiality. Computer search terms he used showed he had an interest in murder, violent sex and rape.

An extraordinary element of the case was the way Godden’s body was found. Halliwell led police first to O’Callaghan’s body and then to the field where Godden’s remains were to be unearthed.

But because the officer in charge of the O’Callaghan inquiry, Steve Fulcher, had ignored some of the rules governing the way suspects should be treated, Halliwell initially escaped justice over Godden’s murder.

Memory said he believed Fulcher had acted honestly, but was wrong in law. “There is a distinct possibility that Becky would still be in the ground now if it wasn’t for those actions. I’m utterly convinced she’d still be there now. Nobody knew she was missing. They believed she had started a new life in Bristol.”

Asked if Fulcher, who resigned after facing disciplinary action and was now working as a consultant in Somalia, had been vindicated, he said: “It’s very difficult for me to say. He made a very honestly held decision. However it was wrong in law.”

Halliwell will be sentenced at Bristol crown court on Friday.