Stephen Quigley, 26, from Neilston, was found in the grounds of the Royal Alexandra Hospital off Craw Road at around 7.20am on Wednesday.
Police are still treating his death as unexplained.
But enquiries made by Police Scotland’s major investigation team have established Mr Quigley may have been at an address in the area of Bourock Square in the Auchenback area of Barrhead, on Tuesday. It is believed he may have travelled from there to the hospital in a blue Ford Focus at around 5pm that day.
Detectives are asking anyone in either of these areas, who may have seen the blue Ford Focus around this time on Tuesday evening, to get in touch.
They are also keen to view any relevant dash-cam footage that may have captured a vehicle of this description being driven erratically, or any suspicious activity.
Police said an assault later took place at around 2am on Wednesday in Bourock Square, and a 26-year-old man was seriously injured.
Officers want to speak to anyone who may have witnessed anything suspicious in the area at this time.
Detective Chief Inspector Scott McCallum said: "Stephen's family are being provided with support by specialist officers and have been updated. Our thoughts are with them at this time.
"We would urge anyone who believes they may have seen Stephen prior to him being found on Wednesday morning to please get in touch as soon as possible.
"In particular, we are keen to speak to anyone who may have been driving near to the hospital, or in the area of Bourock Square, Barrhead, around 5pm on Tuesday, 23 March, and saw a blue Ford Focus.
"Anyone who can help our enquiries is asked to call 101, quoting reference number 0409 of 24 March. Alternatively, Crimestoppers can be called anonymously on 0800 555 111."
Mr Quigley is described as being white, around 5ft 10in tall, and of slim build with short dark hair.
At the time of his death, he was wearing a blue hooded top that was zipped up, black tracksuit bottoms and black trainers.
A major incident public portal has also been set up online to allow the public to submit information directly to the major investigation team.
This allows witnesses to quickly and easily send information, reports, images and video footage directly to police as part of a national online reporting tool for major incidents. It gives the public access to various forms in the event of a terror attack, major disaster or high profile incident such as a murder.
It can be accessed HERE.
A message from the Editor:
Thank you for reading this article. We're more reliant on your support than ever as the shift in consumer habits brought about by coronavirus impacts our advertisers.
If you haven't already, please consider supporting our trusted, fact-checked journalism by taking out a digital subscription at https://www.scotsman.com/subscriptions.