Note: If necessary, delete an existing greeting (limit of 2 greetings) to make room for the new greeting: Tap the Menu key, tap Delete greetings, tap the check box next to the desired greeting, and then tap Delete. Tap the Record icon to record your greeting. Tap the Stop recording icon to stop recording. Start by tapping on the phone icon found at your iPhone’s home screen. Go to the setting section on your iPhone. Now, you can go to the keypad on your phone and then type the number #404 and then call so that you are able to turn off voicemail on iPhone. How do you delete your voicemail greeting on iPhone? Tap Voicemail, then tap a message. Do any of the following: Play the message: Tap . Messages are saved until you delete them or your carrier erases them. Share the message: Tap . Delete the message: Tap . Why can’t I change my greeting on my iPhone?
2. Access the dial pad. You'll need to bring up the Phone app's dial pad, as if you were going to call an actual person. To do this, tap the dial pad icon near the bottom of the screen.

There are two tiers of paid accounts, giving you ad-free access, the ability to “Slydial” multiple people simultaneously, and other premium features. For most people, the free version is perfectly adequate.
Slydial is an app that is available for both iOS and Android. Free accounts are available but they require you to listen to an ad before making your call.