We derive a new initial value representation for semiclassical time-correlation functions. This derivation combines the initial value formalism developed by Miller with the stationary phase analysis of integrals over endpoint velocities developed by Xiao and Coker [J. Chem. Phys. 102, 496 (1995)] and more recently extended by Bonella, Ciccotti, and Coker [Molec. Phys. 62, 1203 (1996)]. As a result, the determination of the classical paths within the correlation function does not require "root" searches; furthermore, the thermal density matrix within this function weights the initial and not the final positions of these paths. To prevent the correlation function from being not a smooth function of time, a semiclassical phase index similar to the Maslov index is introduced. A simple numerical example is provided and possible criticisms of our approach are discussed.