Adaptive filters are applicable to many signal processing tasks including acoustic echo cancellation, beamforming, and more. Adaptive filters are typically controlled using algorithms such as least-mean squares (LMS), recursive least squares (RLS), or Kalman filter updates. Such models are often applied in the frequency domain, assume frequency independent processing, and do not exploit higher-order frequency dependencies, for simplicity. Recent work on meta-adaptive filters, however, has shown that we can control filter adaptation using neural networks without manual derivation, motivating new work to exploit such information. In this work, we present higher-order meta-adaptive filters, a key improvement to meta-adaptive filters that incorporates higher-order frequency dependencies. We demonstrate our approach on acoustic echo cancellation and develop a family of filters that yield multi-dB improvements over competitive baselines, and are at least an order-of-magnitude less complex. Moreover, we show our improvements hold with or without a downstream speech enhancer.