David Hunter, policy and communications officer at Acevo, said she was moving in order to make the most of her marketing background and pursue more "front-line work".
Scott-Spicer's post will be covered by Acevo staff members Abby Wright, who will become director for membership, and Nick Aldridge, who will take on her remaining duties.
At Rainer, Scott-Spicer will oversee marketing, communications, fundraising and policy work. "It will be very satisfying to apply Acevo's policy work on public service delivery and funding reform at Rainer," Scott-Spicer said in a statement.
Joyce Moseley, chief executive at Rainer, said that Scott-Spicer would bring "an enormous wealth of experience" gained in the public and third sectors.
"We want to make a difference to even more young people across the country, and Gail will play a big part in that," Moseley said.
Stephen Bubb, chief executive of Acevo, said that Scott-Spicer had made a significant contribution to the organisation's growth.