Lisa Squire and Dame Diana Johnson, the Labour MP for Hull North, called for referrals to programmes for offenders subject to Sexual Harm Prevention Orders and Sexual Risk Orders.
Dame Diana said the call came after police told her Miss Squire’s murderer would not have been referred under the current system for earlier offences including voyeurism.
She added if offenders like Pawel Relowicz were referred for lower level ‘non-contact’ crimes it could stop their behaviour escalating and becoming violent and deadly.
The MP and mother of Miss Squire wrote in a letter to Home Secretary Priti Patel that more needed to be done to stop such “heinous crimes” happening in the first place.
It comes as Dame Diana tabled amendments to the Policing Bill requiring mandatory referrals in non-contact cases but they were rejected by Safeguarding Minister Victoria Atkins.
The MP is now to call for the Home Office to mount a campaign encouraging the public to report low level offences and to re-table the previous amendments with some changes.
Dame Diana said the referral amendments would now call for a presumption that referrals should be made unless there was good reason not to.
She added it and a campaign were needed so women, girls and the public could feel safe and more confident that reports of lower level offences would be taken seriously.
The MP said: “When I spoke to the police about what happened with Libby, it became very clear that the man who raped and murdered her had been prowling the streets for many months.
Dame Diana and Mrs Squire told the Home Secretary that under the system for handling non-contact offences, “very little” could have been done to address Relowicz’s behaviour.
Relowicz is serving 27 years in prison after attacking Miss Squire and dumping her body in a river. He committed a string of sex offences against women before killing the 21-year-old.