The ability to add a new user/organization from the ticket would add too much clutter and complexity to the page. However we can see the benefit of being able to see the user's organization in the ticket header. We will take this under consideration.

After reviewing this further internally, we are concerned that displaying the organization in the ticket header would push the limits of the header size, especially on mobile devices. Instead, we are currently planning to add the organization to the User header, which can be quickly accessed by clicking on the user's email address in the ticket header.