One of the main reasons Pyre does not have a New Game Plus, Supergiant Games says, is because its campaign already has many threads of replayability. But, after you complete the campaign the first time, you'll be brought back to the main menu, where it just gives you the option to "Load Checkpoint" or "Continue Campaign," which is right above the Versus Mode.
So how do you start a new game in Pyre? It's a bit more convoluted than it has to be, but it's definitely possible. Instead of hitting "Continue Campaign," go to the "Settings" menu and then hit "Save Slots." See the image below:
After that, simply interact with one of the unused save slots, and it will take you back to the main menu where the top option is "Start Campaign." We wish you all successful Rites, and may the Scribes smile down upon you all.