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I think you can recover your voicemail from your iPhone. Do you have any backup files from iTunes or iCloud? If yes, I think you can recover your voice mail from them. If not, you can recover deleted voicemail from your iPhone directly with iPhone Data Recovery software.
If you accidentally deleted voicemail, you can restore your phone with backup on iTunes or iCloud. But if you don't have backup, you might need to use a data recovery app that enables you to recover deleted voicemail iPhone. Some apps can help you do recovery task without using backup but directly retrieving deleted files from the device.
To retrieve a deleted voicemail, please first check your Voicemail app directly to undeleted it, go to check your iTunes or iCloud backup to see whether it contains the one you need. If so, you can get it back by restoring iTunes/iCloud backup.
Otherwise, you need to recover deleted voicemail by using a third-party iOS data recovery tool. Among various recovery apps, I strongly recommend Joyoshare iPhone Data Recovery, which is powerful and secure.
You can get the free version to have a try. And here is the detailed guide:
Generally speaking, there are 2 methods to retrieve a deleted voicemail on you iPhone 5.
First, Check ''Deleted Messages'' Items to see whether those deleted voicemails still exist.
Secondly, You can try TunesKit iPhone Data Recovery, which is able to rescue you from any iOS data disaster by restoring lost data from iPhone including photos, reminders, videos, contacts, iMessages, WhatsApp, notes, etc. This data recovry tool provides three smart recovey modes, recover from iPhone directly, recover from iTunes Backup and from iCloud Backup.
It is easy to operate following these three steps: connect your iPhone to the computer, click the "Scan" button and waiting for the scanning progress and finally select and recover deleted voicemail from your iPhone.
You cannot retrieve a deleted voicemail with that software.
The voicemail is never saved to the phone nor to the iTunes or iCloud backup as data. The voicemail remains on the network servers and once deleted cannot be recovered.
This could be wrong. Voice mail can be sycned to iCloud or iTunes backup if you enabled the options. If not, they will stay on the local storage of iphone. theoretically, they can be recovered with apps or software like this one suggests. however, it really depends on what kind of situation it is.