A variant of compression optical coherence elastography for mapping relative tissue stiffness is reported. Unlike conventionally discussed displacement-based (DB) elastorgaphy, in which the decrease in the cross-correlation is a negative factor causing errors in mapping displacement and strain fields, we propose to intentionally use the difference in the correlation stability (CS) for deformed tissue regions with different stiffness. We compare the parameter ranges (in terms of noise-to-signal ratio and strain) in which the conventional DB- and CS-approaches are operable. It is shown that the CS approach has such advantages as significantly wider operability region in terms of strain and is more tolerant to noises. This is favorable for freehand implementation of this approach. Examples of simulated and real CS-based elastographic OCT images are given.