Long-lived hyperfine coherences in rare-earth-ion doped (REID) crystals provide controlled phase evolution windows for metrology. We use this resource to propose an experimental protocol for accessing strain shifts of ground state hyperfine transitions in non-Kramers REID materials, a quantity that is difficult to access directly with conventional spectroscopic methods. Starting from the crystal field Hamiltonian, we relate the response to strain-dependent effective quadrupole and Zeeman tensors, including changes in electronic wave functions, virtual electronic admixtures, and the bare nuclear quadrupole interaction. The protocol optically prepares a selected hyperfine class, uses a phase-controlled rf $\pi /2$ pulse to create the sensing coherence, and applies synchronized dynamical decoupling pulses during the phase evolution interval so that a coherent ac strain drive accumulates phase rather than averaging away. The accumulated phase is then retrieved by Raman heterodyne readout. The resulting framework provides a route to understanding strain-induced hyperfine shifts in non-Kramers REID systems.