VoLTE comes to the AT&T Samsung Galaxy S4 mini thanks to OTA update5
AT&T is requiring you to have a Wi-Fi connection when downloading and installing the update. After all, depending on the version of Android currently employed on your phone, the file could range in size from 102MB to a whopping 595MB. And make sure that you have a sufficient charge on your phone. Considering the possible size of the update, you might want to have the battery charged to at least 50% before you proceed.
As with most OTA updates, AT&T is sending this out in waves. That means you might not be one of the lucky ones who receives it today. In fact, it might be several days before it arrives. You can try to coax it our of your phone by going to Settings > More > About > Software updates.
source: AT&T via AndroidCentral