Sudesh Amman was shot dead by police around a minute after launching an attack on two bystanders in Streatham High Road, south London, on Sunday. A third person was injured by flying glass during the gunfire.
Police said he began stabbing the victims about 60 seconds after entering a shop, where he stole a knife and removed the packaging as he ran from a staff member.
The 20-year-old, who was jailed for possessing and distributing terrorist documents in December 2018, was freed from prison less than a fortnight ago and had been staying at a bail hostel in nearby .
A former prisoner who spent time with Amman in Belmarsh high-security prison told The Times: "The guy was definitely insane and he never hid his intentions, so it's crazy how he even got out of jail."
The man said Amman wanted to copy the murder of Ms Cox, who was killed in her Batley and Spen constituency in 2016, and told him "the only way to get these filthy kafirs (non-believers) out of Syria is to take out MPs like that white guy did with the lady in 2016," The Times reported.
The former inmate told the newspaper that Amman "wanted to do something real, something organised like the (IRA) used to do".
IS supporter Amman, who at the time of his sentencing was 18 and living in Harrow, smirked as he was jailed for three years and four months at the Old Bailey in December 2018.
The Metropolitan Police said he was released from prison on January 23, 2020.
It comes as new Tory MP for Bishop Auckland Dehenna Davison admitted Ms Cox's murder nearly made her quit politics altogether.
Speaking to The House magazine Ms Davison, who is one of the youngest MPs at just 27 and who first stood for Parliament in Hull North in 2015, said: “If there’s one thing that made me question whether or not it was something I wanted to do, it was what happened to Jo Cox."
She said: “She was so young and so vibrant, and seemingly so harmless like she wouldn’t hurt a fly; I had I never had the pleasure of meeting her but from what I heard, she was a really wonderful, kind, generous woman.
“So for someone to do that, to an MP who also wasn’t super high profile, that was the biggest concern. It was a real moment of confusion, concern, sadness. I remember calling my mother and saying ‘what do I do?' Because part of me wanted to just step away and do something where I wouldn’t be at risk, but part of me thought...is it better to actually stand up and prove to them that they’re not going to put off people going into politics? And eventually I settled on the latter.”
Meanwhile, analysis by foreign policy think tank, the Henry Jackson Society, suggests that more than 10 terrorists are due to be released from prison in 2020.
Yesterday Justice Secretary Robert Buckland said emergency legislation is needed to make sure terrorists are not released automatically from prison half way through their sentence.
Mr Buckland said offenders will not be considered for release until they have served two-thirds of their sentence and that no terrorists will be released before the end of their full custodial term unless the Parole Board agrees.
The Parole Board would be "strengthened" to deal more effectively with the risks that terrorists pose and steps would be taken to introduce the plans "as soon as possible", he said.
The Government will also consider making new legislation to ensure that extremists are more closely monitored on release and will review whether the current maximum sentences for terrorist offences are sufficient.
Legal experts raised concerns about plans to apply the new rules to prisoners who have already been sentenced, with former independent reviewer of terrorism legislation Lord Carlile telling BBC2's Newsnight: "I think it may have gone too far.
"The decision to lengthen the sentences of people who have already been sentenced, and therefore expected to be serving half the sentence the judge imposed upon them, may be in breach of the law.
"It is certainly going to be challenged."
Scotland Yard said armed officers were following Amman on foot as part of a "proactive counter-terrorism surveillance operation" when the incident took place.
The three victims were taken by ambulance to south London hospitals.
One man, in his 40s, is in a serious but stable condition, police said.
A woman in her 50s, a teacher at a nearby school, had non-life threatening injuries and has been discharged from hospital.
Police said a second woman, in her 20s, who suffered minor injuries believed to have been caused by glass following the discharge of a police firearm, has also left hospital.
Investigations continue with officers also searching two addresses in south London and Bishop's Stortford, Hertfordshire.