Voicemails/ Voice messages should be always important compare to calls and Messages for you personally or in your business just, might be useful for later use. Normally Voicemail message allot more space, you have best option for move or save iPhone voicemails to another iPhone or on Mac/ PC. For the security reason you can take backup of voicemails to your Mac or PC.
Before number transfer to another carrier, you must tack backup of your voicemails to iCloud or on Mac/ PC. There is no guarantee about all voicemails transferred to another carrier (Verizon, AT&T, and Sprint).
In case you lost or deleted voicemails and want back on iPhone, So you can restore it from old iCloud backup.
Alternate Ways on Move or save iPhone voicemails to another new iPhone, Mac/ PC
All Voicemails transfer to new iPhone or Voicemails restore from Old backup
Transfer voice mail to another iPhone is quite easy and very useful. You can do that securely.
Step 1: First take backup of your iPhone from settings. Settings > iCloud > Backup under the storage details.
iCloud backup stored in your iCloud account, and accessible with same apple ID activated your iPhone.
Now you next step should be restore taken backup on new device
Time on setup new device, you need to choose last taken backup from list of all backups. All data automatically transferred to new iPhone (All contacts, Calendars, Voicemails).
Transfer or Move/ Copy all Voicemails to Mac/ PC
To move or save iPhone voicemails to Desktop or Mac OS X, You have 3.5 mm male to male audio pin cable.
Audacity is a Great free program for record Voicemails to Mac or Windows. Download and install on your PC (Windows 8, Windows 7)/ Mac (OS X Yosemite, EI Capitan).
Connect male to male 3.5mm jack cable on iPhone and Mac/PC’s microphone port.
Start recording on Audacity software from Mac and iOS device directly. Give use your feedback on which way you prefer first to move or save iPhone voicemails.