Homomorphic Encryption (HE) enables Private Inference (PI) in Machine Learning as a Service (MLaaS), protecting both client inputs and server-side neural network (NN) parameters. Existing PI techniques are predominantly implemented as either HE-based fire-and-forget methods or MPC-based interactive methods. Recent HE-based PI systems improve the accuracy--performance trade-off via a layer-wise scheme and parameter switching, yet remain bottlenecked by fire-and-forget execution in which the server alone performs costly ciphertext management (e.g., bootstrapping and scheme/parameter conversions). We present HEPIC, an HE-based PI system that explores a different design point by leveraging client interventions for ciphertext managements. In a sense, HEPIC shares a common ground with MPC-based PI of being interactive with the client, but differs in that the client only intervenes for ciphertext managements required in HE operations. Because ciphertext management has identical semantics on the client and the server, HEPIC lets developers decide where and how often to execute it, enabling fine-grained trade-offs among computation, communication, and ciphertext configuration. HEPIC makes such execution practical by overlapping client re-encryption, server computation, and communication via dependency-aware pipelining and streaming-based transfers. We further enhance the performance with a cache-aware task allocator (CATA) and a cost-aware client intervention scheduler (CACIS) to exploit ciphertext-level parallelism and to mitigate stalls under client-server performance disparity. Our evaluation shows that HEPIC achieves up to 2.20--41.93× speedup over state-of-the-art fire-and-forget HE-based PI, while maintaining zero loss in inference accuracy.