If you had a generic voicemail greeting and recorded a personalized voicemail greeting you can change to the custom voicemail greeting by saving your custom voicemail recording. You should first listen to your recording by playing it from the menu and can then choose to keep the custom greeting or revert to the default one. Don’t forget to confirm your changes by tapping “Save” when done. How do I change my voicemail message on Samsung?How do I delete my voicemail greeting on Android?How do I change the default voicemail on my Android?How do I manage my voicemail on Android?How do I change my voicemail greeting?How do you use voicemail on Samsung?How do I remove a name from my voicemail greeting?How do I turn off my voicemail message?How do I turn off voicemail on Samsung?How does voicemail work on Android?What is my voicemail PIN?What is the best voicemail app for Android?How do you unlock voicemail on Android?Is there a voicemail app for Android?Does Samsung have a voicemail app? On Android devices above Android 5 (Lollipop), open the Phone app. Then, press and hold “1” to call your voicemail. Now, enter your PIN and press “#”. Press “*” for the menu. Press “4” to change settings. Press “1” to change your greeting. Tap the three dots (upper right corner of screen) Tap “settings” Tap “voicemail” Tap “advanced settings” Tap “setup. Tap “voicemail number. Enter your 10-digit phone number and Tap “OK. Tap the home key to return to the main menu. Open your device’s Phone app ​. At the top right, tap More . Tap Settings. Voicemail. You can: Change what handles your voicemails: Tap Advanced Service. Set up your voice mailbox: Tap Advanced Setup. Change your notification settings: Tap Notifications. Open the Google Voice app . At the top left, tap Menu Settings. In the Voicemail section, tap Voicemail greeting. Next to the greeting you want to use, tap More Set as active.
4. Toggle Cellular Service on and off. You can turn off and then turn on Cellular Service from Settings or Control Center. Launch Settings → Cellular → Toggle off Cellular Data → After 30 seconds, toggle it back on.

Website: https://www.reddit.com/r/apple/comments/3lm2eh/difficulty_changing_voicemail_greeting_in_ios_9/
You will be asked to delete your current Voicemail greeting. Click on the ‘Yes’ button that appears on the screen if you want to remove that. You can save your new recorded Voicemail greeting or can use the default recorded option. Wait and hear the new Voicemail greeting message that the instructions speak out. If you want to keep that recording as your Voicemail greeting, then save it. Or you can re-record the message to get a different greeting.