The young man and a friend were first attacked by a group of about five youths outside the Terrace Bar in the John Street area of the Co Tyrone town, in the early hours of New Year's Eve.
A short time later, as Jason tried to get a taxi home, he was attacked again - this time outside the Weigh Inn on Kevlin Road - after being followed by at least three of the original group of youths.
He was struck so hard that he was knocked to the ground and officers believe that may have been the blow that killed him. A post-mortem examination has shown that he died from head injuries.
Jason was helped into the taxi by two men who had not been part of either fracas, but who may have witnessed the second incident.
While he managed to return to his friend's house in Emyvale, Co Monaghan, he was found dead in his bed there after midday on Monday.
If you were involved, come forward to detectives in Omagh and we will interview you to establish the facts.
DCI Richard Harkness
The PSNI particularly want to speak to a man in his 20s who is about 5ft 4ins tall and of stocky build, with ginger or blond hair, and urged that individual to contact them to assist with inquiries.
He was involved in the incident outside the Weigh Inn and was wearing a white T-shirt at the time.
Anyone else with information about what happened is also asked to come forward, especially local taxi drivers or other passing motorists and the middle-aged men who went to Jason's aid.
"I need to establish what took place and what was said between the assailants and the victim," Detective Chief Inspector Richard Harkness, who is leading the investigation, said.
"Don't delay, come forward. There are two sides to every story. I want to establish the facts."
CCTV footage will also be examined and gardaí are assisting with the investigation.
DCI Harkness added that the incident was consistent with so-called 'one-punch' fatalities that have claimed the lives of 20 people in Northern Ireland since 2004.
"This is an assault that had no apparent motive other than young men out on a night out and fuelled by alcohol," he said.
The victim was understood to be the youngest of three sons and a student at Sligo IT College.