There are several carrier restrictions with the
visual voicemail feature. If you switch to a new cellular plan after updating your iPhone to iOS, your carrier settings may need to be updated as well to work with the new iOS platform.
On your iPhone 6s, navigate to Settings-> General-> About section, then check next to Carrier for any pending carrier settings updates. Install any pending updates and see how much of a difference they make.
Once the carrier settings update has been successfully completed, reboot your iPhone 6s
and test visual voicemail to check whether it now works properly.