The growing demand for intelligent applications beyond the network edge, coupled with the need for sustainable operation, are driving the seamless integration of deep learning (DL) algorithms into energy-limited, and even energy-harvesting end-devices. However, the stochastic nature of ambient energy sources often results in insufficient harvesting rates, failing to meet the energy requirements for inference and causing significant performance degradation in energy-agnostic systems. To address this problem, we consider an on-device adaptive inference system equipped with an energy-harvester and finite-capacity energy storage. We then allow the device to reduce the run-time execution cost on-demand, by either switching between differently-sized neural networks, referred to as multi-model selection (MMS), or by enabling earlier predictions at intermediate layers, called early exiting (EE). The model to be employed, or the exit point is then dynamically chosen based on the energy storage and harvesting process states. We also study the efficacy of integrating the prediction confidence into the decision-making process. We derive a principled policy with theoretical guarantees for confidence-aware and -agnostic controllers. Moreover, in multi-exit networks, we study the advantages of taking decisions incrementally, exit-by-exit, by designing a lightweight reinforcement learning-based controller. Experimental results show that, as the rate of the ambient energy increases, energy- and confidence-aware control schemes show approximately 5% improvement in accuracy compared to their energy-aware confidence-agnostic counterparts. Incremental approaches achieve even higher accuracy, particularly when the energy storage capacity is limited relative to the energy consumption of the inference model.