The hydrodynamic properties of free-standing smectic films close to a second order smectic-A (SmA) nematic (N) phase transition are investigated. By using an extended McMillan model, the profiles of the smectic elastic constants are computed at different temperatures. We show that the amplitude of smectic fluctuations has a strong dependence on the film temperature. By considering the critical behavior of viscoelastic coefficients of thin films, we show that the typical underdamped relaxation of the smectic order is replaced by an overdamped behavior close to the bulk transition temperature.