Eight former and serving officers have been called to give evidence at the hearings.
South Yorkshire Police and Crime Commissioner Alan Billings said the sum was "substantially less" than required.
The force, which has a legal duty to support its staff, said that as of January more than £12m had been paid out on legal representation.
The total spend had now risen to "considerably more than that", it added.
Dr Billings described the financial situation for the force as "grave" and said taxpayers in the region would "have to bear the additional cost".
He said it would ultimately impact on the police budget and the force's ability to provide policing and crime services.
Dr Billings said he would be calling an emergency meeting.
Ninety-six Liverpool fans died after crushing at the FA Cup semi final match between Liverpool and Nottingham Forest held at Hillsborough in 1989.
Fresh inquests into the deaths were ordered after the original accidental death verdicts were quashed in 2012 following an independent report.
The inquests are expected to continue for another year.
Dr Billings said: "The special grant application I submitted was for the full amount, which leaves us the shortfall to pay.
"My office now has to undertake a financial analysis to establish the size of the shortfall and its impact on our budget."
"While I recognise that the Home Secretary's decision was a matter of discretion, I am extremely disappointed by both the timing of the announcement and the amount of funding granted."
Dr Billings said he would submit another application to fund "the further legal costs we will incur during 2015/16".
A Home Office spokesman said: "On 27 March the home secretary agreed to provide a special grant to the South Yorkshire police and crime commissioner in relation to legal costs incurred by the inquests into the Hillsborough tragedy.
"The amount is up to £5.1m in 2013/14 and £5.6m in 2014/15."
Special grants can be provided to police forces facing unexpected or exceptional costs exceeding 1% of its annual budget.

The 22-year-old ex-West Ham youngster made only six appearances for the Tykes after joining last summer.
He previously played for the Hatters on loan from West Ham, scoring three goals in 11 matches during a loan in 2014-15.
The son of former Newcastle midfielder Rob Lee joins his brother Olly, who is already at Kenilworth Road.
"I'm looking forward to playing with Olly. He was a big factor in me coming here. Having been here and known all the lads, I'm really looking forward to it," he said.
"Olly and I spoke about it and we realised it would be a great experience to play together and something we'd enjoy. It's something special for the family."
Lee makes the opposite move to midfielder Cameron McGeehan, who left Bedfordshire to join Barnsley last month.
Find all the latest football transfers on our dedicated page.
South Yorkshire Police has been awarded £10.7m to cover legal fees incurred from the new Hillsborough inquests.
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Luton Town have made forward Elliot Lee their seventh addition of the summer after signing him on a two-year deal following his release by Barnsley.