In its just-released report on the legislation, the Justice and Electoral Committee said it did not support relaxing the rules further.
The committee pointed to advice from the Director of Mental Health. The director said media could play a positive role in suicide awareness, but there was also strong evidence that reporting of suicide methods could "popularise" those methods.
The committee said the bill sought to narrow reporting restrictions on the details most likely to lead to copy-cat behaviour, including the site of a self-inflicted death if it hinted at the method of suicide.
Not only did the committee reject calls to soften the rules, it introduced a new restriction on publishing details about suicides.
The amendment banned media from reporting any information about a suicide, including the location, which could suggest the method of death.
"We consider these amendments would achieve a balance between providing for the freedom and role of the media in suicide reporting, and the need to prohibit the publication of the problematic elements of suicide reporting," the committee's report said.
At present, no information about a death that appeared to be self-inflicted could be made public unless the coroner granted an exemption.
The law changes would also remove a requirement for coroners to hold an inquest if a person died in official care or custody.
Instead, the coroner would have discretion over whether they started an inquest - a change which the committee said would reduce unnecessary investigations.
The Chief Coroner welcomed the increased discretion.
But Labour's justice spokeswoman Jacinda Ardern said the amendment was a backwards step.
She said the removal of mandatory inquests was particularly troubling when it came to deaths of children in Child Youth and Family (CYF) care or in prison.
"To not take the opportunity to make it a requirement for those who are referenced - such as Government agencies - to follow up coroners' recommendations is puzzling."
During public consultation on the bill, MPs were lobbied by victims of the Carterton air balloon crash to make changes around coroners' obligations after a death.
Parents Vivienne and Alan Still said they waited six days to see their body of their daughter Alexis in the 2012 disaster, which killed 11 people.
They wanted coroners to be required to advise victims' families of their rights at a death scene.
The committee acknowledged the Still's "frustration and anguish" but said it would not agree to the amendment.
"We consider that the difficulties encountered by Mr and Mrs Still were specific to the individual case, were unlikely to have broader implications, and could not be remedied by changes to the legislation."
Suicide reporting rules
NOW:
At present, no information about a death which appears to be self-inflicted can be made public - unless the coroner agrees - until the coronial inquiry is completed.
If a coroner decides that a death is suicide, the only information that can be made public is the person's name, job and address and the fact that the death was self-inflicted.
PROPOSED:
Under the proposed reforms, the media will be able to report a death as a "suspected suicide" before a coroner's inquiry is completed, if the facts support that conclusion.
The Chief Coroner will be able to grant an exemption for a suicide to be reported on if satisfied that the risk of copycat behaviour is small and is outweighed by the public interest.
FURTHER CHANGES:
A committee now says publication restrictions should also apply to any information - including the location - which could hint at the method of a suicide.
