Use call answering rules to determine what happens when a caller reaches your voice mail. For example, if you have automatic replies turned on, then you could set up your rule to transfer the caller to a different number. If you have multiple rules, you can specify the order in which they are applied.
I can access my voicemail from my phone just fine. I’m just cheap and used Hangout/Google Voice to save my minutes.

On the iPhone, go into voicemail and tap "Greeting". Record a few seconds of something and tap stop. Before you tap save, use iFuntastic or SFTP to navigate to /var/root/Library/Voicemail on your iPhone. You'll see a file called "Greeting. amr". This is the audio you just recorded. Delete that file and replace it with your custom message which should then be renamed to "Greeting.amr". Tap "Play" on the iPhone to verify your message works. If it plays, tap "Save" and your custom message will be uploaded to AT&T's servers.
To set up voicemail on your iPhone, tap Phone on your main screen. Tap Voicemail in the bottom menu. Tap the Set Up Now button. Choose a passcode to secure your voicemail. Use the default greeting or record your own. Finally, tap Save to set your outgoing voicemail message.