Completing this task was simple. First, I got to the attach file window using the following path
Evolution → Compose Message → 1st <no name> → 1st <no name> → 2nd <no name> → 1st <no name> → 1st <no name> → 3rd <no name> → Attach → [Take Action]
Then I attached the file using the following path:
Evolution → Instert Attachment → 1st <no name> → 1st <no name> → 1st <no name> → 1st <no name> → 3rd <no name> → 2nd <no name> → 1st <no name> → 1st <no name> → Files → choose a file → [Take Action]
Although I find uploading files less annoying than downloading files, I still think this dialogue box could be simpler.