I successfully (so far at least) went to Python 3.6.5 using the guide, so massive thanks to @turboc for putting it all in one place. The only issue I had wasn’t from this guide, but I thought it might help someone doing what I did while following the guide.
After following the guide and restarting HA, I got a python error importing the pip.locations function. When I decided to upgrade Python to the latest release (and deal with any issues along the way), I also followed the pip notification that pip 10.0.0 was out and I was running 9.0.1, and upgraded to the latest version there too. That was the issue, so I had to downgrade to pip 9.0.3 (using pip3 install pip=9.0.3). More details in this thread (huge thanks to @pemontto for that post).
Hope that helps somebody…
Dox