Defence Force chief Lieutenant General Rhys Jones described Corporal Grant as a "competent and likeable" soldier, who saw himself as "an SAS soldier from the start to the finish".
His family said he "had absolute faith in his friends and colleagues and what he was doing in Afghanistan".
Corporal Grant died after climbing onto the roof of a building next to the British Council
compound to co-ordinate fire and provide other information to the Afghan Crisis Response Unit, where he was hit with a rifle or light machine round which entered the armhole of his body armour and pierced his torso, hitting his heart.
Despite his injuries, he clung to life for some time and was resuscitated at least once. However, he died during the helicopter flight to Bagram Military Hospital.
Corporal Grant had spent a third of his 21-year military career as an infantryman and a further third with the Royal Engineers.
The rest had been with the SAS, an outfit he had "the most personal identity with", General Jones said.
His death is the first SAS fatality in Afghanistan in four deployments since 2001.
