FFT Phase interpretation of input signal with non-integer number of cycles in FFT window
There is a circular waveform discontinuity at sample 0 of an FFT input (to sample N-1), if it isn’t exactly integer periodic in aperture.  However, if the waveform is continuous at sample N/2, then the phase can be measured at that point.

You can measure phase at the halfway point by doing an FFTshift, or by flipping the phase of every odd numbered FFT output bin.  If you know, or can estimate the frequency, then if you estimate the phase at sample N/2, you can use the frequency and phase at a known point to compute the phase at any other point in time.  Including at the beginning of the original sample data window.

This answer is for even N Lengths of FFT.  (Dawg may have a better answer for odd lengths.)