Disparity in phase calculation between STFT and Hilbert transform
Without seeing your data, it's hard to tell whether you have contributions of closer frequencies or not.  That is certainly possible.  So, my suggestion to you would be this: try out your two different methods on pure sine waves where you know that there are no closer frequencies.  In other words, generate some dummy data which is just one single frequency.  If your two methods give the exact same answer on that data, then maybe it is in fact the contributions of closer frequencies that are messing up your original calculation.  But if the two methods do not give the exact same answer, then you likely have an error in your method.