Derivative cloud service providers use nesting to provision virtual computational entities (VCE) within VCEs, e.g., containers runtimes within virtual machines. As part of resource management and ensuring application performance, migration of nested containers is an important and useful mechanism. Checkpoint Restore In Userspace (CRIU) is the dominant method for migration, used by Docker and other container technologies. While CRIU works well for container migration from host to host, it suffers from significant increase in resource requirements in nested setups. The overheads are primarily due to the high network virtualization overhead in nested environments. While techniques such as SR-IOV can mitigate the overheads, they require additional hardware features and tight coupling of network endpoints. Based on our insights of network virtualization being the main bottleneck, we present Portkey - a software-based solution for efficient nested container migration that significantly reduces CPU utilization at both the source and destination hosts. Our solution relies on interposing a layer that directly coordinates network IO from within a virtual machine with the hypervisor. A new set of hypercalls provide this interfacing along with a control loop that minimizes the hypercall path usage. Extensive evaluation of our solution shows that Portkey reduces CPU usage by up to 75% and 82% at the source and destination hosts, respectively.