How to get Siri to dial my contact called "voicemail"?
I know your question states: Other than changing the name of the contact, what can I do to get Siri to dial this contact?, but you don't clarify why you don't want to do it this way?

I'm not aware of another way to do this, and in my case changing the contact name works flawlessly. I just created a contact called "Company Name Voicemail" and Siri has no problems whatsoever.

I just say, "Hey Siri, call..." and she just dials the number and I go from there.