If you recently restored or updated your iPhone, your network setting may have been affected and may be causing “iPhone voicemail error try again later.” Please reset your network settings. Go to Settings > General > Reset > Reset Network Settings Enter your password. Press Reset network setting in red font. The phone will automatically reset it. After it restarts, your phone then tries voicemail again.
As a matter of fact, some reports that have explored the decline of voicemail say the service is really only used by two types of people anymore: older parents and business people. But that still doesn’t lessen the annoyance of the receiver when they have to listen to the voicemail from mom and pop or the used car sales guy.

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If you are recording a voicemail message for business, be sure you include your name and your company's name, so people know who they are calling and were they are reaching them. If you do not include the business name, they may think they have mistakenly called you at your personal number. You will also want to make sure to give callers the option to speak to someone who is currently available, an operator or other representative, by pressing a certain key (check with IT personnel to see what this might be).