By having the alternate greeting pre-recorded and assigned, you can change your voicemail greeting on the fly. Recorded Name: Set this option to play a recording of your name instead of a greeting. “Leave a Message” Prompt: Check this field to play the following message after your greeting: Please leave your message after the tone.
But now I'm going to go ahead and record. So press record sorry I missed your call. Please leave your voicemail after the beep and I'll call you back. Thank you. Press stop and now. I could press play and then if I don't like what I hear, I could press record one more time and when I'm satisfied, I could press save on the top right and it's going to save my greeting and that's all I have to do to set a new voicemail on my iPhone. You could always come back to this page. Press record one more time and replace your voicemail here. You're greeting here, and you should always use someone else's phone to call to see how it sounds from a different device.

1. Start the Phone app and, if you don't see the numeric keypad, tap "Keypad" at the bottom of the screen.
The default, generic voicemail greeting that comes with your iPhone is fine but also rather bland and boring. You can personalize the voicemail greeting on iPhone by recording your own custom version of it. This is a tutorial on how to create a custom iPhone Voicemail greeting.