The transverse polarization of forward Lambda hyperons produced in high-energy p-A collisions is expected to display an extremum at a transverse momentum around the saturation scale. This was first observed within the context of the McLerran-Venugopalan model which has ail x-independent saturation scale. The extremum arises due to the k(t)-odd nature of the polarization-dependent fragmentation function, which probes approximately the derivative of the dipole scattering amplitude. The amplitude changes most strongly around the saturation Scale. resulting in a peak in the polarization. We find that the observation also extends to the more realistic case in which the saturation scale Q(s) is x-dependent. Since a range of x and therefore Q(s) values is probed at a given transverse momentum and rapidity. this result is a priori not expected. Moreover, the measurement of Lambda polarization over a range of x(F) values actually provides a direct probe of the x-dependence of the Saturation scale. This novel feature is demonstrated for typical LHC kinematics and for several phenomenological models of the dipole scattering amplitude. We show that although the measurement will be challenging, it may be feasible at LHC. The Situation at RHIC is not favorable, because the peak will likely be at too low transverse momentum of the Lambda to be a trustworthy measure of the Saturation scale. (C) 2008 Elsevier B.V. All rights reserved.