The 46-year-old was found with a head injury in Monsal Street, Thurnscoe, last Thursday.
Police were called shortly after 4.40pm and Mr Garner was taken to hospital, where he remained in critical condition before he died on Sunday.
Four men have been arrested on suspicion of murder, and a 28-year-old and 31-year-old remain in police custody.
One of these men was initially arrested on suspicion of assault along with a 41-year-old man, who has since been bailed pending further enquiries.
A 29-year-old man has also been bailed.
Detective Inspector Rob Platts, who is overseeing the murder investigation, said: “We have specially trained officers currently supporting Mr Garner’s family, who are understandably devastated at losing Stafford.
“They have questions, quite rightly, about what happened to their loved one and how he came to be gravely injured.
“I appreciate that while many of us have been staying indoors as a result of government guidance last week, there may still be people out there yet to make contact with officers who know something about what happened to Mr Garner.
“If you live locally and saw Mr Garner in the days before he was found, we’d like to hear from you. If you have had a conversation with someone who is holding information about this incident, we want to hear from you too.
“No matter how small or insignificant you feel your piece of information may be, it could help us build a better understanding of how Mr Garner came to be fatally injured.”
A post mortem examination carried out earlier this week provided the preliminary cause of death as a head injury.
This is, however, subject to further investigations.
If you have information that could help police, call 101 quoting incident number 494 of April 2, or access the MIPP site here: https://mipp.police.uk/operation/14SY20B03-PO1
Independent charity Crimestoppers can be contacted via its website - www.crimestoppers-uk.org - or by calling its UK contact centre on 0800 555 111.