A personal voicemail greeting is an easy way to prevent callers from hearing a generic greeting message. It ensures that people know they've reached the correct person, and makes them feel more comfortable leaving a voicemail. To change your voicemail greeting on an iPhone, you'll need to follow a few simple steps.
However, some cellular service providers allow users to access and update their voicemail by calling their own numbers from working landlines or phones on other wireless networks, so users whose accounts include such an option and who have access to alternative service could keep the voicemail greetings updated even when their own mobile phones cannot connect to cellular or internet service.

6 hours agoListen to and manage your U-verse Voicemail messages by phone or online. Dial *98 from your U-verse home phone, dial your home phone number from another phone, or access your voicemail messages online from the Voicemail Settings tab. Press the * key when you hear your greeting. Enter your PIN. Press 4 and follow the prompts.
Verizon only: Unfortunately, if you have set up the voicemail successfully but still unable to change the greetings of Voicemail from the Voicemail tab, then dial *86 and select option 4 to make a change. Step #1: Launch the “Phone” app on your iPhone. Step #2: Tap “Voicemail”. Step #3: Select “Greeting” on the upper-left corner of the screen. Step #4: In case, you are also using eSIM then select the second network to change voicemail greetings. Step #5: Select “Custom”. From here you can record a greeting. Step #6: Tap “Record”.