Dark matter comprised of axion-like particles (ALPs) generated by the realignment mechanism in the post-inflationary scenario leads to primordial isocurvature fluctuations. The power spectrum of these fluctuations is flat for small wave numbers, extending to scales accessible with cosmological surveys. We use the latest measurements of Cosmic Microwave Background (CMB) primary anisotropies (temperature, polarization) together with CMB lensing, Baryonic Acoustic Oscillations (BAO) and Sunyaev Zel'dovich (SZ) cluster counts to measure the amplitude and tilt of the isocurvature component. We find preference for a white-noise isocurvature component in the CMB primary anisotropies; this conclusion is, however, weakened by current large-scale structure (LSS) data. Interpreting the result as a conservative upper limit on the isocurvature component, the combined bound on the ALP mass from all probes is m(a) greater than or similar to 10(-19) eV, with some dependence on how m(a) evolves with temperature. The expected sensitivity of cosmic shear and galaxy clustering from future LSS experiments and CMB lensing suggests improved bounds of m(a) greater than or similar to 10(-18) -10(-13) eV, depending on scale cuts used to avoid non-linearities and the ALP mass-temperature dependence.