EOC media director Claire Hibbitt said institutions needed to 'be more creative in taking account of the needs of women and men when coming up with their policies'.
She cited as good examples one Midlands council that has located a GP surgery in a barber's shop to address men's traditional reticence to visit a doctor; another council has enticed women to take up plumbing apprenticeships by helping with childcare costs.
Luther will drive a consultation process with equal opportunities staff and chief executives at local authorities and NHS trusts, which will help create an EOC 'code of practice'.
It will also push the aims of the bill across the national and relevant trade media.
'This project is a priority for the EOC and makes up a significant part of what is a very important bill,' said Hibbitt.
Described by the EOC as 'the most radical change in equality law for 30 years', the Equality Bill is expected to become law next year.
Luther picked up the business following a pitch against AS Biss & Co and LLM Communications.