The victim, named locally as 42-year-old James Nash, suffered serious head injuries during the attack in Upper Enham on Wednesday afternoon. His death was confirmed on Friday.
His suspected attacker died in a police chase after trying to flee on a motorcycle.
A 40-year-old woman was also assaulted during the incident and suffered minor injuries, Hampshire Police said.
According to the Andover Advertiser, Mr Nash’s mother, Gillian Nash, said on Friday: “Very sadly I have to tell you that James Nash died of his brutal injuries today.
“His father, his sister and I are in a total state of shock and grief.
“We have lost a beautiful, talented son and brother and I know all who knew him would say he was the kindest, most caring person.”
Phil North, leader of Test Valley Borough Council, also confirmed Mr Nash had died of his injuries.
Posting on Facebook, he said: “It has been confirmed this evening that the victim of the Upper Enham shooting on Wednesday, local resident James Nash, has now sadly died.
“This is such devastating news, not just for his family and friends who will obviously miss him terribly, but for the whole community of Enham Alamein where he was a public representative.
“He was such a kind-hearted individual and a proactive Parish Councillor who cared deeply for his community.”
North West Hampshire MP Kit Malthouse described it as “tragic and profoundly sad news”.
“James’s family will be devastated and they are in all our thoughts tonight,” he said in a post on Facebook.
“I know that Hampshire police are working flat out to try and understand what could have sparked this appalling chain of events, and I am grateful for their sensitive and professional handling of this deeply distressing situation.
“James will be deeply missed by his community and the terrible manner of his death will be seared in the memory of everyone in the area. May he Rest In Peace.”
Online council records gave Mr Nash’s address as Holly Tree Cottage on MacCallum Road, where the attack took place at around 3pm.
Mr Nash served as the vice chairman of Enham Alamein parish council.
Mr North added: “As a talented children’s author and illustrator, I was extremely touched last year when one of the dedications in his latest book was to my newborn daughter, Eleanor-Ivy Mae.
“We will always treasure our copy.
“My thoughts tonight though are with James’ family and friends and the people of Enham Alamein.”
Mr North earlier sought to reassure residents of the area, telling them officers were “confident” the deceased rider, 34, was the attacker involved in the “horrific shooting”.
Police have said that while investigations are at an early stage they “do not believe there are any outstanding suspects in this investigation”.
The suspect, who is believed to have initially fled the scene on foot before trying to get away on a motorbike, fatally crashed on an A-road around three miles away from the shooting site after officers gave chase.
He died at the scene and his next-of-kin have been notified as formal identification takes place, said the force.
Hampshire Police said officers “also attended an address in Enham Alamein as part of this investigation”.
A statement added: “Officers were called to an address in MacCallum Road, Upper Enham, at 3.01pm on Wednesday August 5, following a report of an assault on a man.
“The man had been shot and sustained serious head injuries.
“Detectives are continuing to investigate the circumstances of these incidents.
“While inquiries into the incident in Upper Enham are at an early stage, detectives do not believe there are any outstanding suspects in this investigation.”
A Hampshire Constabulary spokeswoman said the force has made a mandatory referral to the Independent Office for Police Conduct (IOPC).