Is there a version of Welch's method that doesn't look for power?
Since an FFT is a linear operator, adding up the complex results of a sequence of FFTs of short windows is the same as doing a single short FFT on the the vector addition of all those short windows.

Note that for signals that are exactly integer periodic in the FFT width (sequential 0% overlapped windows), the vector addition will constructively interfere.  But for signals that are not exact integer periodic in the FFT width, there can be significant destructive interference.  So the result will be highly scalloped, with a narrow high gain bandpass around each FFT bin center, and potentially a lot of signal loss between bin centers.  Sometimes this is useful, sometimes not, but a very different result from a single longer FFT over the entire window set.