Network telemetry, characterized by its efficient push model and high-performance communication protocol (gRPC), offers a new avenue for collecting fine-grained real-time data. Despite its advantages, existing network telemetry systems lack a theoretical basis for setting measurement frequency, struggle to capture informative samples, and face challenges in setting a uniform frequency for multi-metric monitoring. We introduce FineMon, an innovative adaptive network telemetry scheme for precise, fine-grained, multi-metric data monitoring. FineMon leverages a novel Two-sided Frequency Adjustment (TFA) to dynamically adjust the measurement frequency on the Network Management System (NMS) and infrastructure sides. On the NMS side, we provide a theoretical basis for frequency determination, drawing on changes in the rank of multi-metric data to minimize monitoring overhead. On the infrastructure side, we adjust the frequency in real-time to capture significant data fluctuations. We propose a robust Enhanced-Subspace-based Tensor Completion (ESTC) to ensure accurate recovery of fine-grained data, even with noise or outliers. Through extensive experimentation with three real datasets, we demonstrate FineMon's superiority over existing schemes in reduced measurement overhead, enhanced accuracy, and effective capture of intricate temporal features.