An East Riding of Yorkshire Council spokesperson said locals there would be able to find details about who qualifies on their website once their funding scheme is launched.
It comes as the councils received a combined £1,308,450 over and above the Government money for the £150 rebates to distribute as they see fit.
The money is aimed at households not in council tax bands A to D, those who qualify for the rebate, but that councils deem in need of further financial help.
A total of 300 councils have been given a combined £144m in discretionary funding.
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The funding could total up to £40 per households, depending on how councils chose to allocate it.
Government guidance states that it is up to councils to decide how best to use the extra cash to help those outside the scheme struggling financially.
Elsewhere in the country, Wyre Forest District Council is paying households an extra £35, regardless of their property tax bands.
Bromley Council has opted to give out £40 top up payments to households in bands A to D receiving council tax support.
The discretionary funding comes after Chancellor Rishi Sunak unveiled the £150 rebates in February as part of a package aiming to cushion the blow of spiralling living costs on households.
So far Hull City Council has paid the rebates out to 68,465 households, a total of £10.27m, with 21,000 paper application forms currently being processed.
In the East Riding, a total of £12.9m has been paid out to more than 86,000 households the council holds direct debit details for.
Details for a further 12,500 households who do not pay council tax by direct debit are currently being verified because rebates are sent out.
East Riding Council’s spokesperson said: “The council is working on the detail of the discretionary household support funding and is not able to comment further until we have everything fully worked through. As usual, the council’s website will be updated when we launch the scheme to allocate this funding.”
A total of 132,330 East Riding homes out of a total of 159,010 are in council tax bands A to D, around 83 per cent, making them eligible for the £150 rebate.
A total of 121,250 of Hull’s 123,020 homes were in Council Tax Bands A to D as of September 2021, about 99 per cent of the total according to Government figures.