The role of Stormont's special advisors became the subject of controversy in 2011 when Mary McArdle took up the position under Culture Minister Carál Ní Chuilín.
She had been convicted of involvement in the murder of 22-year-old Mary Travers, the daughter of Resident Magistrate Tom Travers, almost 30 years ago.
Ms McArdle vacated the position in March last year following a campaign by the victim's sister.
TUV leader Jim Allister brought the bill before the assembly, so that potential advisors should be vetted in future to prevent a similar appointment.
The Special Advisor's (Civil Service) Bill seeks to ban anyone who has been sentenced to more than five years in prison from taking up the posts in future.
Sinn Féin tried to amend the bill so that candidates with a history of conflict-related offences could be referred to a panel from the Office of First and Deputy First Minister.
But, on Tuesday night, the party's bid failed and the bill passed the consideration stage during a debate in Stormont.
It is right that the primary principle of this bill, that anyone with a serious criminal conviction, of whatever sort, should be prohibited from holding a post of special advisor at the heart and top of government.
TUV leader Jim Allister
Speaking following the debate, Mr Allister said: "I'm encouraged that the amendments of Sinn Féin were so soundly defeated.
"They wanted to exclude, for special provision, (the block on) anyone with what they called a conflict-related offence, whereas crime is crime, murder is murder, no matter in what purpose it is committed.
"Of course it flows out of the scandalous appointment of Mary McArdle, who was convicted for her part in the murder of young Mary Travers and the outrage that created particularly for the Travers family, in the retraumatising of them as victims," the TUV leader continued.
"This bill is about protecting victims from that in future."
Reacting on Wednesday, deputy First Minister Martin McGuinness criticised the SDLP for supporting the TUV bill.
But the SDLP hit back, claiming that Sinn Féin conceded with Mr Allister's argument by acknowledging that someone with a serious criminal conviction could be unsuitable or ineligible in their amendments.
"This is a major concession by Sinn Féin who maintained the opposite stance through the Committee stage of the Bill," SDLP MLA for Newry and Armagh Dominic Bradley said.
During the course of this debate Sinn Féin offered no challenge to the TUV or any other Unionist party. It was left to the SDLP to fulfil that role. Both Sinn Féin contributions were weak and failed to defend their position.
SDLP MLA Dominic Bradley
"The SDLP stood by its amendments through the debate, stood up to, and faced down attacks from both Unionism and Sinn Féin," he added.
He also criticised Finance Minister Sammy Wilson, who is responsible for special advisors, for sitting on the back benches, as well as Alliance party MLAs for changing their stance on the bill.
The SDLP amendment included introducing an appeals process, so anyone being considered for a role with a history of serious offences would be brought before a panel of civil service commissioners.
The amendment was not passed, but a similar mechanism brought by Mr Allister did go through, where commissioners would only approve the candidate if satisfied that there were exceptional circumstances justifying it.
These include, whether the person has shown contrition for the offence to which the serious criminal conviction relates; whether the person has taken all reasonable steps to assist in police investigations relating to the offence as well as taking on board the views of the victims.
SDLP Justice spokesperson, Alban Maginness, paid tribute to Ms Travers for bringing the issue to the forefront.
"It is an inspiration to everybody that one person could generate so much interest and create a campaign that has proven to be so successful, at least thus far," he said.
"The SDLP amendments offered the strongest case for victims as well as being fair to potential appointees and existing incumbents. I regret that the Assembly did not accept all of our amendments but we will continue our rigorous work on this Bill at its future stages."
Leslie Cree, Ulster Unionist Finance and Personnel spokesperson, commented: "The Ulster Unionist Party has been supportive of this Bill from the outset as it seeks to effectively regulate the appointment of special advisers who have serious criminal convictions. We believe that the Bill will result in a fairer, more victim-centred approach to these sensitive appointments."
The bill will now go to further consideration stage in the assembly.