Detectives issued an updated description of the man yesterday in a bid to jog people's memories.
The woman was attacked in the graveyard of the Canongate Kirk sometime between midnight and 2.30am on Wednesday morning, and a police spokesman said that the area was busy with Hogmanay revellers at the time. About 80,000 revellers were taking part in celebrations in the city centre.
Forensic teams have been carrying out a search of the Canongate Kirk grounds and officers are continuing their efforts to trace the man.
Officers have described the suspect as white, in his late teens to early twenties with a slim to medium build. He was said to be wearing a distinctive furry headband, and detectives think this may have made him stand out from the crowds.
Anyone who recognises the description or who may have any other information relevant to the investigation is asked to contact police immediately.
Detective Inspector Bobby Deas said: "Through ongoing enquiries we have obtained a more consistent description of the suspect and detectives are now progressing with the investigation to trace this male.
"Anyone who believes they can help identify him is asked to come forward.
"Similarly, we continue to urge those who remember seeing anything suspicious in or around the Canongate in the early hours of New Year's Day to get in touch."
Meanwhile, Police Scotland is also investigating a sexual attack in Aviemore in the Highlands in the early hours of New Year's Day.
A woman was assaulted at around 1.30am in the area known as the tunnel, near the Vault nightclub.
Two hours later, a man was seriously assaulted in Inverness city centre in a separate attack.