A judge heard that Mohammed Akram had killed while in his teens and was still on licence after serving time for a sex attack on a 16-year-old student.
At the High Court in Edinburgh, Lord Brodie ordered an in-depth investigation into the possible risks Akram, 52, poses to the public.
Such a report could clear the way for a life sentence to be imposed when Akram returns to court in May.
A trial heard how guests were enjoying a New Year party - to see in 2011 - with the four-year-old girl’s mum at a house in Edinburgh when Akram sneaked away to commit the offence.
She gave evidence in a video interview, telling how Akram came into the room pushed her onto a bed and touched her over her clothes.
Akram denied the offence but a jury found him guilty.
After the verdict it was revealed that Akram had been jailed for five years in 1979 for culpable homicide.
In 2003 he was convicted of a sex attack on a 16-year-old student in Alloa. The court heard then that he dragged her around a house in Alloa and made her commit an indecent act.
Akram was jailed for seven years and, in addition, his licence conditions were extended by three years - meaning they still applied at the time of the New Year offence.
When he was accused of molesting the four year old his licence was recalled and he was sent back to prison.
Lord Brodie ruled today that he should remain on remand pending sentence.