Besides creating an audio file in the right format (Adaptive Multi-Rate), you also need to trick the iPhone to upload your new greeting to the AT&T servers:
The importance of voicemail in the customer care sector has not diminished despite the arrival of newer web technologies. VoIP applications let the businesses use voicemail service with flexibility, as it is. The users need not rely on extension terminals when they use VoIP services. A strong internet connection is a prerequisite and the pre-configured phones can be used – both for calling /receiving calls and for automated custom greetings. IVR feature redirect callers to the right virtual extension where their needs can be addressed. This can be done anywhere through a web portal access and an internet connection.

The telephone systems have the feature named as Voicemail. VoIP is the best example of it. When a receiver is unable to pick up the phone, then the caller can leave the voice message on the receiver’s phone. In older telephonic machines, such messages were stored as a voice message directly, and the receiver can hear them later. In modern telephone systems, such info is being stored on the server of the service provider in the user’s mailbox. It is similar to email in which the messages are saved in the form of audio instead of text.
Website: https://www.iphonelife.com/content/how-change-your-voicemail-greeting-your-iphone