The figure for 2017/18, confirmed by children's minister Edward Timpson, represents a £4.2m increase on the £35.8m councils received in 2016/17.
It will be supplemented by £19.1m to support work by charities to help families navigating the changes, making a total of £59.1m - a 26.1 per cent reduction on the £80m provided by government for 2016/17.
Under government plans, all children with existing SEN statements must be assessed and transferred from the old system onto new education, health and care (EHC) plans by April 2018.
Councils will also be legally required to take the views of families into account when deciding on what support to offer.
But there are fears that councils will struggle to be ready in time.
Announcing the funding, Timpson said: "As we enter the final year of the transition, I know there are still challenges to overcome, to ensure that the inspiring work going on in many parts of the country is shared with areas where improvements still need to be made.
"That's why I'm delighted to be able to confirm this additional funding for councils and for the groups playing such a vital role in supporting children with SEND. All children, no matter the obstacles they face, should have the same opportunities for success as any other."
The bulk of the £19.1m earmarked for charities will be used to continue funding the Council for Disabled Children's (CDC) Independent Supporters programme.
The programme will receive £15m in 2017/18 to provide independent advice and support to families around education, health and care (EHC) plans, which replace statements of SEN as part of the reforms.
Latest evaluation of the programme, carried out by the National Development Team for Inclusion (NDTi), found that nine out of 10 families found it "very" or "extremely" useful and 81 per cent of those carrying out the work felt they had been well trained.
CDC director Dame Christine Lenehan said: "Independent Supporters has fast become a valuable resource to help parents of children and young people with SEN navigate the EHC planning process and get involved in annual reviews."
Meanwhile, parent carer forums, of which there is one in each council area, will receive £2.3m over the next two years to help them gain a greater voice in shaping local SEN services.
In addition, £1.8m is to be handed to the charity Contact-a-Family, to further support parent carer forums as well as run a national helpline for families.
Earlier this week the DfE launched a tender process for seven contacts worth a total of £4.8m to support efforts to implement SEN reform.