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Several variational quantum circuit approaches to machine learning have been proposed in recent years, with one promising class of variational algorithms involving tensor networks operating on states resulting from local feature maps. In contrast, a random feature approach known as quantum kitchen sinks provides comparable performance, but leverages non-local feature maps. Here we combine these two approaches by proposing a new circuit ansatz where a tree tensor network coherently processes the non-local feature maps of quantum kitchen sinks, and we run numerical experiments to empirically evaluate the performance of the new ansatz on image classification. From the perspective of classification performance, we find that simply combining quantum kitchen sinks with tensor networks yields no qualitative improvements. However, the addition of feature optimization greatly boosts performance, leading to state-of-the-art quantum circuits for image classification, requiring only shallow circuits and a small number of qubits -- both well within reach of near-term quantum devices.