To efficiently scale large model (LM) training, researchers transition from data parallelism (DP) to hybrid parallelism (HP) on GPU clusters, which frequently experience hardware and software failures. Existing works introduce in-memory checkpointing optimizations that snapshot parameters to device memory for rapid failure recovery. However, these methods introduce severe resource competition between checkpointing and training, which can work under DP but can hardly scale under resource-intensive HP. To ensure low checkpointing overhead for hybrid-parallel training, this paper introduces a distributed in-memory checkpointing system with near-zero in-memory saving overhead. It strives from two aspects to mitigate the on-host resource competition caused by in-memory checkpointing: (1) It introduces Hierarchical Asynchronous Snapshotting Coordination in the checkpoint saving stage. This approach uses three-level asynchronous on-device scheduling to enhance parallelism between snapshotting and training, thereby minimizing snapshotting overhead. (2) It proposes Hybrid In-memory Checkpoint Protection to enhance checkpoint completeness during hardware failures. Unlike methods that require inter-node communications, which may block training under HP, it creates intra-node redundancy with efficient resource utilization, protecting training against hardware failures with minimal overhead. With these methods, this work enables fast restart for failed HP training with Distributed In-memory Checkpoint Loading, bypassing inefficiencies in NFS reads. In our evaluation, we achieve zero in-memory checkpoint saving overhead on Frontier while training Llama-2-34B on 256 MI250X devices (512 GPUs).