Is there proof theory for some more or less usual logics that is outside the scope of structural proof theory (Hilbert/natural deduction/sequent calculi)? E.g. proof theoy for adaptive logic (https://www.springer.com/gp/book/9783319007915) of defeasible logics that resembles logical programming or forward reasoning. What other proof techniques outside structural proof theory are available? Is it possible to classify and generalize them?
This question is concerned with more exotic theories such as ordinal analysis, mentioned in https://en.wikipedia.org/wiki/Proof_theory. I am more concerned with practical logics who are doing their proofs non-structural way.
Or maybe there are logics without proof theory (or any proof system) at all?