Much of what you'll be asked to do, no doubt, relates to 'prep' work in between shows, such as copying scripts or sides for the next auditions, archiving old prompt scritps or scores, and keeping the 'continuity' going so that what is learned from one show is applied to the next. If you're the common factor, then helping new directors or musical directors 'adjust' to the way your company likes to have things done will help avoid problems before they start. Depending on how much staff there is at the company, you may end up overlapping duties with a general manager, such as booking rooms for visiting artistic staff, etc. but I would ask for something extra if that happens, as you'll be doing work outside the usual SM boundaries.
You might be able to negotiate for health benefits, eg, even when not on contract....
Since it's SPT, I'm guessing you'll be damn near an assistant producer in reality....
Art