Labour has written to the home secretary, Priti Patel, calling for an ongoing review into the safety of MPs after the killing of the Conservative backbencher Sir David Amess to be extended to others in public life, including councillors.

Under the proposal by Nick Thomas-Symonds, the shadow home secretary, the review would look into the safety of others such as prominent public servants and journalists.

On Friday leading councillors called for action including a specific law against intimidating public officials and a greater willingness to prosecute those who make threats, with one recounting how he had been sent a death threat because of a temporary traffic light scheme.

Prof Chris Whitty, the chief medical officer for England, has been harassed by anti-vaccine protesters, as have a number of journalists and broadcasters.

The review was set up after Amess was stabbed while holding a constituency surgery in Leigh-on-Sea, Essex. Ali Harbi Ali, 25, was charged with his murder last week.

Thomas-Symonds said: “It is completely unacceptable that people who are in public life, be that councillors, public servants, journalists or broadcasters, should fear for their safety for just doing their jobs. We must do everything possible to stop a repeat of the awful killing of MPs Sir David Amess and Jo Cox and the awful attacks that others have faced.

“Labour is committed to working on a cross-party basis to drive forward the review into the safety of MPs. We now are calling on the home secretary to step up that work, by urgently establishing a part-two of the review, that will assess the security risks and what preventive tools can be made available for others in public life.”

Patel ordered the immediate review of MPs’ security on 15 October, the day Amess was killed, after meeting police and representatives of the security and intelligence agencies, as well as the Commons Speaker, Sir Lindsay Hoyle.

Hoyle also promised to look again at MPs’ security after Amess’s killing, which came five years after Cox, a Labour MP, was murdered by a rightwing terrorist.

Last week Patel told MPs the threat level against them had been formally raised to “substantial”. Making a Commons statement after the killing of Amess, she urged them to take the “change in risk seriously” after a review by the Joint Terrorism Analysis Centre.

In the letter to Patel, Thomas-Symonds stressed that he backed the review into MPs’ security, and that a part-two element “should not need to hamper or slow down the ongoing review into the security of members of parliament, but instead be based on sharing key lessons”.

A Home Office spokesperson said: “The government backs the police to use the full force of the law to go after anyone who intimidates, harasses or abuses elected representatives, including councillors. “