The most natural way to address the tiles is to use a sequence of indices based on the hierarchical construction.

Simon Tatham has written about this, in the context of the original hat tile: https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...

Addressing hierarchically constructed tiling is quite similar to addressing hyperbolic tilings. Both the "hat" tiling and "spectre" tiling already work in HyperRogue. (Spectre is not yet released but pushed to GitHub.)

There is a brief explanation here: https://twitter.com/ZenoRogue/status/1638997769141604360

You might be interested in this https://twitter.com/ZenoRogue/status/1663589972094353418