The pseudorapidity (eta) distributions of charged hadrons are measured using data collected at the highest ever nucleon-nucleon center-of-mass energy of root s(NN) = 5.36 TeV for collisions of lead-lead ions. The data were recorded by the CMS experiment at the LHC in 2022 and correspond to an integrated luminosity of 0.30 +/- 0.03 mu b(-1). Using the CMS silicon pixel detector, the yields of primary charged hadrons produced in the range |eta| < 2.6 are reported. The evolution of the midrapidity particle density as a function of collision centrality is also reported. In the 5% most central collisions, the charged-hadron eta density in the range |eta| < 0.5 is found to be 2032 +/- 91 (syst), with negligible statistical uncertainty. This result is consistent with an extrapolation from nucleus-nucleus collision data at lower center-of-mass energies. Comparisons are made to various Monte Carlo event generators and to previous measurements of lead-lead and xenon-xenon collisions at similar collision energies. These new data detail the dependence of particle production on the collision energy, initial collision geometry, and the size of the colliding nuclei.