And you’re done! Your CenturyLink voicemail is now set up. Whenever you have an unheard message, you'll hear a stuttered dial tone when you pick up your home phone. Take a few minutes to gather your thoughts, even jot down a few notes, and practice before you record. Before you start recording, turn off anything in the background that might cause noise. This will ensure your voice is clear and easy to understand. While clever greetings can be fun, it's worth taking a moment to think about the range of potential callers who may be leaving you voicemail. Consider the tone and image you want to project. Don't worry! If you don't like your recording, you can erase it and re-record as many times as you'd like.
If the problem is with your phone, the factory reset has a good chance of solving it. If not, contact your mobile service provider, to carry out some troubleshooting to help resolve the problem from their end.

Tap “Phone” on your iPhone’s “Home” screen and tap “Voicemail” to access your voice-mail settings. Tap “Greeting” and select “Default” if you wish to use the default greeting for your voice mails. Click “Custom” if you want to create a personalized greeting, and tap “Record” to say your greeting.
Another worth considering trick is to remove all the old voicemails that are consuming the storage and creating multiple dilemmas like this. To delete the voicemail here are the steps, Open the Phone app. Tap on Voicemail at the bottom tab. Then tap on Edit at the top-right screen. Select All the Voicemails and at last tap Delete.