Large language models rely on inference-time context including rationales and instructions to improve task performance. Prior work attempts to internalize such benefits via parameter-efficient fine-tuning, and also studies inference-time representation interventions, but these approaches typically update the base model or learn fixed directions tied to global attributes. As a result, it remains unclear how to capture and reuse the internal effect of a helpful context that is specific to a given query. We take a representation-level view and define a context imprint as the hidden-state displacement between context-conditioned and query-only forward passes. We propose HintSteer, which trains a lightweight query-conditioned predictor to estimate the imprint from the query representation and injects it into a frozen model at inference time, approximating the missing context effect without explicitly providing context. Experiments on mathematical reasoning and writing demonstrate the effectiveness of imprint-based steering under no-context inference and indicate that both imprint magnitude and predictability influence the resulting gains.