Toni
We have in the Bendigo collection an Egan (made in Sydney I think) , that was used by Eric Rothacker, from Bridgewater near Bendigo, on Gentleman John to win the 1956 Interdominion final in Sydney.
It has the support at the back of the seat that you describe.
We also have a really nice hickory Royal Speed made in Adelaide. It's the more well known style where the seat is supported on the rear crossbar.