Still, if you are unable to record a custom voicemail greeting on iPhone, then its time where the carrier comes in the picture then for their assistance and get it fixed, the custom voicemail not recording issue.
01-21-2015 05:29 PM. If you call your number from another phone, when the voicemail kicks on hit the # button. It'll ask you for your password. Enter it, and then you'll have access to the Main Menu to change your Voicemail greeting. View solution in original post.

It only take a few minutes to set up your Voicemail account. Once setup is complete, your Voicemail will be ready to receive messages. Call 123 to connect to your voicemail. If prompted for a password the first time you access your account, use the last four digits of your mobile number. When prompted, create a password which can be any 4 to 7 digit code. (A good password is one that’s easy for you to remember, but hard for others to guess) When prompted, record your greeting and name. Your voicemail is now ready to use. Call *86 to connect to your voicemail. If prompted for a password the first time you access your new account, use the last four digits of your mobile number. When prompted, create a password which can be any 4 to 7 digit code. (A good password is one that’s easy for you to remember, but hard for others to guess) When prompted, record your greeting and name. Your voicemail is now ready to use. Accessing your Voicemail Call 123 to connect to your voicemail Press 1 to listen to unread messages If you wish to replay a message, press 1 Press # to skip to the next message Press 7 to delete a message, or press 9 to save for the next 30 days At any time, press * twice to return to the main menu To change your password, press 5 from the main menu Call *86 from your phone Press 1 to play your messages Press # to skip to the next message Press 7 to delete a message, or press 9 to save To return to the main menu at any time, press * To enable or disable voicemail notifications, from the main menu press 4 for personal options, followed by 1 To change your password, press 4 for personal options, 1 for Administrative Options, then 1 again Checking Voicemail from a Landline Call 1-805-637-7249 Then when prompted, enter your 10-digit mobile number. You can skip the greeting by pressing the * key. When you’re prompted, enter your password. And finally, just follow the menu prompts to check your messages. Call your cell phone from the landline Let it ring till it goes to voicemail As soon as you hear the greeting, press # When you’re prompted, enter your password just follow the menu prompts to check your messages.
I can access my voicemail from my phone just fine. I’m just cheap and used Hangout/Google Voice to save my minutes.