Use the non-thresholded version of that linear classifier output as one additional feature-dimension over which you learn a decision tree. Then wrap this whole thing up as a system of boosted trees (that is, with more short trees added if needed).
One of the reasons why it works so well, is that it plays to their strengths:
(i) Decision trees have a hard time fitting linear functions (they have to stair-step a lot, therefore need many internal nodes) and
(ii) linear functions are terrible where equi-label regions have a recursively partitioned structure.
In the decision tree building process the first cut would usually be on the synthetic linear feature added, which would earn it the linear classifier accuracy right away, leaving the DT algorithm to work on the part where the linear classifier is struggling. This idea is not that different from boosting.
One could also consider different (random) rotations of the data to form a forest of trees build using steps above, but was usually not necessary. Or rotate the axes so that all are orthogonal to the linear classifier learned.
One place were DT struggle is when the features themselves are very (column) sparse, not many places to place the cut.
Oblique Decision trees, Model Trees. (M5 Trees for example), Logistic Model Trees (LMT) or Hierarchical Mixture of Experts (HME).
I mention restricted oblique trees in passing. In my experience, oblique trees tend to add considerable complexity, the others more so. Of course this will depend on dataset.
The merit of what I used is in its simplicity. Any simple ML library would have a linear classifier and a tree learner.
Super easy to implement, train, maintain, debug. One to two person team can handle this fine.
Could you explain what "equi-label regions having a partitioned structure" mean?
Consider connected regions in the domain that have the same label. Much like countries on a political map. The situation where this has a short description in terms of recursive subdivision of space, is what I am calling a partitioned structure. It's really rather tautological.