Efficient FFT (or SFFT) for adjacent parts of signal with overlap?
One method might be to split your length $N$ overlapping FFTs up, so that you compute the $\frac{N}{2}$-point FFT of each block of samples that come in. Then, in order to get the $N$-point FFT that you want, you can combine the last two $\frac{N}{2}$-point results with appropriate twiddle factors.

This would technically do what you want, but in most cases (especially on contemporary PC-like platforms with good FFT libraries), it's not going to be appreciably (if at all) faster than just doing the overlapping $N$-point FFTs.