If you go to Tools>System>Presentation>Show>Assignemnt you will see a screen where it seems to let you specify which fields appear in the More grey box. BUT ......I have altered this on my system but it does not make any difference so I think that you cannot alter what appears here.
Go to Presentation/Forms/Assignment and open the form designer.
You will notice that the window is split.
Whatever you have on the left will be presented on the service call form. Whatever you have left and has value it will presented in the gray area.
For example if you move "Assignment from Person" and "Assignment from Workgroup" to the left, these will be presented on the main service call form and you won't need to press the "More" button to edit them.
Note however that this will affect ALL items with assignments (workorders/service calls etc) and not only service calls.