If you aren’t on the same network as the person you want to call, the next-best option is to use an app.
Once you have your message, you need to actually record it. The exact process varies depending on whether you’re using a cell phone or office phone, but here’s the basic process: Press the voicemail button, or press and hold 1 on most cell phones. Enter your password. Record your message. Listen to the message you just recorded. Follow the prompts to save your message. Following Up on Voicemail Messages

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.
Also, carriers have a limit to how long they'll let you record a voice mail. Test it with your phone and a friend before you trust this method.