Update: VoLTE worked on the Verizon SIM after powering off and on with the SIM installed. No such luck with the AT&T SIM. That tells me that it has something to do with SIM provisioning, since clearly VoLTE works. I will go to AT&T tomorrow and ask them what gives.
It looks like it's not just me. I'll call AT&T again tonight and see what happens.
Does the Verizon iP6 work with VoLTE on ATT?
Update 2: So this afternoon I noticed that VoLTE stopped working on my "Verizon" A1549 AT&T iPhone 6 (but was still working fine on Verizon on my SIM-free A1586 iPhone 6). I head to the AT&T store, and after nearly an hour of "troubleshooting" including giving me a new SIM card, and still not getting VoLTE, the rep called tech support. She gave the tech support rep the IMEI of my A1549 iPhone and the tech updated my account with it (I'm thinking that the chat rep replaced it with the IMEI of my SIM-free model yesterday). That did the trick.
For kicks, I put the SIM in my SIM-free. At first it still dropped down to 4G for calls, and continued to do so when I put it back into my A1549 model.
But then I powered down both and restarted. First, VoLTE came back to my A1549 iPhone, and it actually briefly worked on my A1586. But then after that, it dropped back down to 4G with the next call, and naturally it did the same on my A1549 iPhone 6, though I was able to VoLTE back on that one by resetting network settings and doing a soft reset.
Bottom line: This seems to be something on AT&T's side, and it appears to be linked to the goofy way that AT&T has provisioned VoLTE. Apparently both your SIM and IMEI need to match up on AT&T's system, and if it doesn't, it appears to sometimes unprovision VoLTE on your account entirely until an AT&T tech manually fixes it on their system.
They have been a GSM carrier for what, 15 years now, and they still can't figure out how to just provision everything on the SIM?
Also, I occasionally get a "your SIM card sent a text message" on my phone. What I notice is that VoLTE works before I get that message. Once it does, then VoLTE stops working. That has happened on both the A1549 and A1586. I'm now wondering if all this has to do with the carrier settings. Perhaps AT&T needs to issue an updated carrier setting for the A1586 that is separate and distinct from the A1549.