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Cr Kate Wilson won support from most other councillors when, at Tuesday's full council meeting, she asked for a staff report on the "cost and implications" of installing a second weighbridge at the landfill.
The report was expected to be considered during 2019-20 annual plan deliberations in May.
Cr Wilson - the chairwoman of the council's infrastructure services and networks committee - told the meeting the issue had been raised again during the first round of annual plan meetings last month.
That led to fresh discussions about the idea at a meeting of council committee chairs, deputy chairs and senior staff.
Cr Wilson said questions remained about the fairness of the charging regime at the landfill, and the best approach was to seek a fresh report from staff to inform later debate.
It would also help start a "conversation" with the community about the merits of weight-based charging for rubbish, she believed.
The only councillor to speak against the idea was Cr Lee Vandervis, who suggested the landfill's existing weighbridge should be sufficient.
It was supposed to be reserved for commercial operators and large truckloads of waste, but Cr Vandervis argued investigating - or installing - a second weighbridge would be a waste of staff time and money.
"If they want to put someone over a weighbridge, they already can."
Mayor Dave Cull said if it was a waste of money, the staff report would show that before any decisions were made.
The issue had previously been considered in 2014, when councillors decided not to install a second weighbridge - expected to cost more than $100,000 - after public consultation and debate.
The move followed a public outcry about the cost of deliveries to the landfill, sparked by a fee increase and the end of booth operators' ability to use their discretion when applying charges.
They had previously assessed the amount of rubbish being dumped before applying a charge, but were then told to charge only by vehicle size or trailer-load.
A study at the time had indicated 63% of all landfill customers would be better off if a second weighbridge was added.