Straight Talk network changes require a new iPhone?

This one is really hard to answer since Apple's documentation essentially has a try it with your carrier and see approach rather than a clear list of which models do and do not.

Assuming that not everybody on the Internet is lying about everything they say, several sites found with a general web search say that the 6 and 6+ and later models support VoLTE. Of course you have to make sure it is activated as indicated in the link I provided earlier.

Normally on the forums here we are encouraged to rely upon Apple documentation for this kind of thing, but in this case there isn't any. Not even model information (which is hard to find for older models on Apple's site) use the term VoLTE.