I don't think you can change that behavior. I haven't seen anything to indicate you can do that.
Having said that, I don't use my provider's voicemail. I use YouMail
which allows you to have specific greetings for specific callers and if the caller is listed in your YouMail contacts list, their name is listed in the SMS that follows. At least that would give your user more information than the provider's method. Just a thought.