How do I transfer voicemails from old iPhone 4S to new iPhone 5S?

I have an iPhone 4S running iOS 8.1.2. I used iTunes to make a backup of the 4S.

I just got an iPhone 5S (running iOS 8.1.2), and I restored the 4S backup onto the 5S. None of the voicemails on the 4S were restored onto the 5S. I entered my voicemail password on the 5S and it didn't download any voicemails. Apparently AT&T doesn't keep voicemails on their servers after you download them.

All the voicemails are still saved on the 4S. Is there any way to transfer those voicemails onto the 5S?