This article about the restoration of an Atwater Kent 206 describes (among other things!) a method for making a new voice coil/cone (look about half way down):
http://ludens.cl/Radiohis/ak206/ak206.html
(It seems implied to me that he fashioned a new spider out of card stock)
While his restoration wouldn't bear up to scrutiny of a concourse-type inspection, you certainly can't fault this guy's ingenuity. The lengths that he goes to to revive this old set are nothing short of inspirational!
Since you have all the other bits and pieces of the speaker, perhaps a similar method could be improvised...? He had access to a lathe, but maybe a somewhat less precise "former" could be attempted with a vertical fixed dowel rod and a concentric circle marked on the work table as a guide for the outer edge of the cone?