Interpolation of estimated frequency between fft bins: what about phase? I am developing an additive re-synthesier, which involves identifying partials from FFT frames.

Partial identification requires amplitude peaks.  The simplest way to do this is to just look at the amplitude of each bin, but this of course can miss partials that happen to lay between bins, and there energy is spread over the two adjacent bins.

I understand there are various interpolation methods (of varying complexity) to estimate the actual amplitude peak and frequency... however, if taking such an approach, how to estimate the phase offset?  Is it even possible?
If you do an fftshift before the FFT (to center the phase reference in the data frame), you can use the same interpolation methods on the real (even) and imaginary (positive odd) components in the complex FFT result, and thus get interpolated phase estimates (referenced to the original data frame’s center) between FFT result bins.