Casper is a process-based asynchronous progress model for MPI one-sided communication on multi- and many-core architectures. The one-sided communication is not truly one-sided in most MPI implementations: the target process still relies on software progress to complete incoming operations. Casper allows the user to specify an arbitrary number of cores dedicated to background ghost processes and transparently redirects the RMA operations to ghost processes by utilizing the PMPI redirection and MPI-3 shared-memory technologies. Although Casper benefits applications that suffer from lack of asynchronous progress, the operation redirection design might not support complex multiphase applications effectively, which often involve dynamically changing communication density and computing workloads. In this paper, we present an adaptive mechanism in Casper to address the limitation of static asynchronous progress in multiphase applications. We exploit two adaptive strategies, a user-guided strategy and a fully transparent and automatic strategy based on self-profiling and prediction, to dynamically reconfigure the asynchronous progress in Casper according to real-time performance characteristics during multiphase execution. We evaluate the adaptive approaches in both microbenchmarks and a real quantum chemistry application suite, NWChem, on the Cray XC30 supercomputer and an Intel Omni-Path cluster.