Once you have your new iPhone in hand and you are learning how to use it, one of the first things you may want to do is record a voicemail greeting. You have the option of using a default one that tells the person what number they have called, but you may elect for a more personal option.
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.

Standard greeting with phone number: "At the tone, please record your message to [phone number]." Standard greeting with your name: "At the tone, please record your message to [name]." Record a personalized greeting, up to 3 minutes long. A common template you can follow is: "You've reached [name]. We can't get to the phone right now, but if
Part 2: How to Change and Record your Voicemail Greetings on iPhone. Your voicemail message let people know they've reached the right person and reminds them of leaving a message after the beep. So if you get a new iPhone, you have no voicemail message. Or maybe you just haven't updated your voicemail greeting.