A non-monotonic behavior of the velocity gradient of a test particle revolving around a rapidly rotating black hole in the locally non-rotating frame of reference is known as the Aschenbach effect. This effect can serve as a distinguishing signature of rapidly rotating black holes, being potentially useful for the measurements of the astrophysical black hole spins. This paper is the generalization of our previous research to the motion of spinning particles around a rotating black hole with non-zero cosmological constant. We show that both the particle's spin s and the cosmological constant Lambda modify the critical value of the black hole spin ac, for which the Aschenbach effect can be observed; ac can increase or decrease depending on the signs of s and Lambda. We also found that the particle's spin s can mimic the effect of the cosmological constant Lambda for a given ac, causing thus a discrepancy in the measurements of s, Lambda and ac in the Aschenbach effect.