Reconstructing time domain signal with Hanning window
You should be using overlap.  That ought to take care of the "oscillating" and make it unnecessary to undo the Hanning window.

The effect you are seeing comes about because the window values at the edges get very close to (and may even be) zero.  The window edges won't line up perfectly with the signal after you've done your processing in the frequency domain, so dividing by those very small values will cause the peaks you see.

In your place, I would first make sure that I could do the windowed FFT/IFFT and recover the original signal.  When you can do that without getting the "oscillation" you can put your processing back in.