Website: https://www.reddit.com/r/iphone/comments/3xuac5/cant_change_voicemail_greeting/
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Before we dive into the troubleshooting steps, make sure that you have Visual Voicemail set up on your iPhone. Open the Phone app on your iPhone and tap Voicemail in the bottom right-hand corner of the screen. If you see “To retrieve a voicemail first set a password and greeting” on the screen as well as a button that says Set Up Now, then Visual Voicemail has not been set up on your iPhone.
Open the Phone app . Press and hold “1” to call your voicemail. Enter your PIN and press “#”. Press “*” for the menu. Press “4” to change settings. Press “1” to change your greeting. Follow the recorded instructions. How do I change my voicemail message? From the Home screen, tap the Phone icon . Tap Voicemail then tap Greeting (upper-left). Greeting is located in the top-left corner of screen. Tap Custom to record a greeting. Enabled when a checkmark is present. Tap Record to begin recording the custom greeting message. Tap Stop to end recording then tap Save. Call *86 (*VM) from the handset or the 10-digit mobile number from another phone. Follow the instructions for choosing the language options then press the # key to confirm the choice. When prompted, enter the 4-7digit password then press the # key. When prompted, say the signature (or name) then press the # key. Open your Android’s Phone app. It usually looks like a phone receiver at the bottom of the home screen. Press and hold the 1 key on the keypad. Tap Add number. Tap Service. Tap My carrier. Tap Setup. Tap Voicemail number. Type your mobile phone number and tap OK.