Robert Pride-McLeod, 49, was found dead in a cellar at the hostel on Robert Street on 21 January 2007.
An inquest recorded an open verdict into his death but post-mortem tests found he had suffered a head injury.
Detectives, who first reopened the case in 2015, said they are determined to find those responsible for his death.
Mr Pride-McLeod, who had heart problems, was made homeless nine days before his death.
Three people were arrested during the initial investigation but were released without charge.
It is thought Mr Pride-McLeod was assaulted shortly before he died.
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Det Insp Mark Pearson of North Yorkshire Police said he believes there are still people in the Harrogate community that hold vital information about what happened but have not yet come forward.
"Someone will know what happened in the final moments of Mr Pride-McLeod's life and I'd urge them to get in touch," he said.
Mr Pride-McLeod was made homeless on 12 January 2007 and was given a room on the first floor of the hostel.
His room was found empty three days later and he was found dead later that week.
No-one has reported seeing him since he was helped to his room by a security guard, Mr Pearson said.
The house where Mr Pride-McLeod was found is no longer a homeless hostel but at the time there were seven rooms and a flat which were occupied.
Detectives have renewed an appeal for information about the death of a Harrogate man in a homeless hostel 10 years ago.