We demonstrate a hybrid approach to autonomous flipper control, focusing on a fusion of hard-coded and learned knowledge. The result is a sample-efficient and modifiable control structure that can be used in conjunction with a mapping/navigation stack. The backbone of the control policy is formulated as a state machine whose states define various flipper action templates and local control behaviors. It is also used as an interface that facilitates the gathering of demonstrations to train the transitions of the state machine. We propose a soft-differentiable state machine neural network that mitigates the shortcomings of its naively implemented counterpart and improves over a multi-layer perceptron baseline in the task of state-transition classification. We show that by training on several minutes of user-gathered demonstrations in simulation, our approach is capable of a zero-shot domain transfer to a wide range of obstacles on a similar real robotic platform. Our results show a considerable increase in performance over a previous competing approach in several essential criteria. A subset of this work was successfully used in the Defense Advanced Research Projects Agency (DARPA) Subterranean Challenge to alleviate the operator of manual flipper control. We autonomously traversed stairs and other obstacles, improving map coverage.