What exactly is the effect of a Hann Window on the FFT output?
See the following: http://www.dtic.mil/dtic/tr/fulltext/u2/a381695.pdf  especially p.631 Figs. 17+18.  As noted by the author, a 4 to 1 overlap using a Hann or Hamming window allows for proper reconstruction.

Also see: F. J. Harris, “On the use of Merged, Overlapped and Windowed FFT’s to Generate Synthetic Time Series Data with a Specified Power Spectrum,” Proc. IEEE 16th Asilomar Conf., 1982, pp. 316-321.