Michael Freshwater, 49, was found at a property in Westridge Road on 29 April.
The photo of Kevin Suika, 23, from London, is being broadcast nationally on BBC's Crimewatch programme in a bid by Hampshire Constabulary to trace him.
Crimestoppers is offering a Â£10,000 reward leading to the arrest and conviction of Mr Freshwater's killers.
Twelve people have so far been arrested in connection with the investigation.
Detectives describe Mr Suika as black, 5ft 7in-tall, of proportionate build, with a tattoo on his left arm that reads "work like a slave, eat like a king".
He has links to the Holloway and Bromley areas of London and Kent.
One man, aged 26, has pleaded guilty to perverting the course of justice. Another, aged 21, is being held on remand, charged with conspiracy to supply class A drugs.
A 16-year-old boy from Kent, arrested in May on suspicion of murder and conspiracy to supply class A drugs, has been re-bailed until 6 July.
A 30-year-old woman from Southampton, arrested in May on suspicion of perverting the course of justice, has been re-bailed until 28 July.
Last month, Mr Freshwater's partner, Jenny Downey, made an emotional appeal for help in tracking down his killers.
She said: "I feel so lost without him and the pain never ends. I still can't accept he's gone."
Hampshire Constabulary said it is believed there was a dispute at the property before Mr Freshwater's body was discovered.
Police investigating the death of a man found stabbed to death in his Southampton home have released a photo of a man they want to speak to.