Updating the voicemail message with your own greeting allows you to let callers know they’ve reached the correct person; the process is fast and easy. How to Change the Outgoing Voicemail Message on IPhone (iOS 9 or later) Tap the phone icon (the same one you use to make a call).
I called into my voicemail, and tried to change the greeting to no avail. After a few attempts, I realized that I had had an "extended absence" greeting in effect. Once I switched to a "custom" greeting, I was able to record and activate a custom greeting.

How to record or change your Android voicemail greeting in 10 simple steps 1. Turn on your phone and launch the Phone app. 2. Open the dial pad. 3. Call your voicemail. 4. Enter your voicemail password. 5. Go to your voicemail settings. 6. Access your greeting or recorded name settings. 7. Edit your voicemail greeting.
If you’ve already created a custom voicemail on your iPhone, you can tap Play to see how your voicemail sounds. If you’re interested in creating a new message, tap the Custom button. From here, you can start recording your message by tapping Record and tapping Stop whenever you’re finished. Once you’re finished recording, tap on Play to make sure the message comes across as you’d like and hit Save (in the upper right) to save it as your voicemail.