We present two approaches to speckle tracking for optical coherence tomography (OCT)-based elastography, one appropriate for small speckle motions and the other for large, rapid speckle motions. Both approaches have certain advantages over traditional cross-correlation based motion algorithms. We apply our algorithms to quantifying the strain response of a mechanically inhomogeneous, bi-layered polyvinyl alcohol tissue phantom that is subjected to either small or large dynamic compressive forces while being imaged with a spectral domain OCT system. In both the small and large deformation scenarios, the algorithms performed well, clearly identifying the two mechanically disparate regions of the phantom. The stiffness ratio between the two regions was estimated to be the same for the two scenarios and both estimates agreed with the expected stiffness ratio based on earlier mechanical testing. No single numerical approach is appropriate for all cases and the experimental conditions dictate the proper choice of speckle shift algorithm for OCT-based elastography studies.