Drupal 7 Views Question - How to pass 2 arguments
I think I know the answer.

I suppose the path to your view without any arguments is "organization".

So first you need to add the contextual filter "department" to the "attachment"-display too. This filter needs to be in first position. 

Then: 

When filter value is not available > check "Provide default value" > type= Raw value from url> Path component= "2" > save this filter.

Next:

Still inside "attachment"-display open up contextual filter "Content: Title" and change Display a summary > Base path to "organization/%/department/%/staff-listing".

Hope this helps