The 31-year-old victim, who has not been named, remains in critical conditions nearly six weeks after he fell from the second floor of the property in Bath Street.
West Midlands Police released CCTV footage on Friday showing a group of men leaving the flats immediately after the incident at around 7pm on Monday 17 August.
“CCTV enquiries show a group of four men leaving the block of flats straight after the incident and we are urgently appealing to trace them as we believe they could hold vital information about what happened,” the force said in a statement.
Two men, aged 25 and 33, were initially arrested at the scene on suspicion of attempted murder, but have both been released without charge.
Police launched an earlier appeal on 18 August asking for help from anyone who may have witnessed the fall.
Sergeant Megan Stokes said: “I am asking for anyone who saw the moments before, during or after the fall to get in touch with us.
“Whatever you witnessed may seem like very minor details, but they could really help us piece together how the man came about falling off the balcony and sustaining life-threatening injuries.”
Anyone with information is asked to contact police via live chat, by calling 101, or calling Crimestoppers with the crime reference number 20CV/196209L/20.