The escaped militant has been identified as Naveed alias Abu Hanzulla, who was arrested from south Kashmir in 2014.
There are reports that Naveed has decamped with the rifle of one of the injured policemen.
A major manhunt has been started to nab him.
A senior police official said they were investigating how the militants got the information that Naveed was being taken to the hospital.
Police said two cops were injured and "one of them very critical".
According to eyewitness accounts, the militants opened fire as soon as Jhutt, along with five other prisoners and the police team, got off the vehicle outside the Out Patient Department of the 70-year-old hospital.
Two Indian police officers have been killed in the attack on Tuesday.
The Kaka Sarai area, where the SMHS hospital is located, has been cordoned off.
Head constable Mushtaq Ahmad was killed while another policeman later succumbed to his injuries in Srinagar's main hospital.
Sources added that a police escort brought Ahmed to the hospital for a medical check-up.
Following the attack, police sounded an alert in the Srinagar district while as police has been asked to intensify the searches in all the entry and exit points in the city. "So far he is untraceable", said an official.
Jutt is a former commander in the Lashkar-e-Taiba militant group.
Sources told DH it was a pre-planned attack to help Naveed flee from the police custody when he was brought to the hospital for a check-up.