This paper introduces Locally Adaptive Neural Context Estimation (LANCE), a novel extension for overfitted image compression (OIC) frameworks like Cool-Chic. While traditional OIC methods rely on lightweight autoregressive networks with globally signaled parameters, they struggle with non-stationary image statistics. LANCE addresses this by incorporating a forward-signaled spatial hyperprior that enables regional adaptation of the entropy model. To minimize overhead, we employ a predictive coding scheme that combines a static Median Edge Detector (MED) with a lightweight learned context model. Experiments demonstrate that LANCE achieves BD-rate reductions of 1.40% on the Kodak dataset and 1.97% on CLIC 2020 over Cool-Chic 4.0 at the high end of our decoder complexity range of 606-1481 MAC/pixel. At the low end of the complexity range, we outperform Cool-Chic 4.0 by 2.41% and 2.99% on Kodak and CLIC, respectively. Qualitative analysis reveals that the learned spatial hyperprior effectively segments image regions into areas of similar image statistics, providing an automated, content-aware adaptation layer.