Alan Bloomer, 27, was killed after being hit by two cars in Lyndhurst.
His silver Ford Focus had hit the central reservation on the eastbound carriageway near Hangersley Hill, at 6.41pm last Friday.
The vehicle then continued along the A31 before stopping by the Rufus Stone services.
Mr Bloomer, of Bellevue Road, Southampton, then left the vehicle and crossed to the westbound side of the carriageway.
He collided with two vehicles and was pronounced dead at the scene.
Mr Bloomer’s brother, John, paid tribute to him on behalf of the family.
In a statement shared by Hampshire constabulary, the tribute said: ‘This was a huge shock to everyone.
‘Our family are devastated that Alan has gone.
‘He probably didn't know how much he would be missed.
‘He was deep down a lovely person, but has struggled with life over time.
‘To anyone over the years who was kind to my brother, I am eternally grateful.
‘I pray he’s at peace.
‘I want him to NEVER be forgotten!’
Anyone with information about the incident is asked to contact Hampshire police by calling 101, quoting Operation Gale and the reference number 44220018123.
People can also submit information via an online form here.