Has there been any research on time-windowed spectral estimation using more than one window function at the same time? In the Welch's method, a signal is divided into overlapping segments, multiplied with a window function, and transformed to frequency domain via fourier transform.

Why does the method use only one window function? Suppose I would use (say) ten window functions to produce ten spectral estimates. By averaging the estimates, my SNR would be higher than by using just one of the windows?
Congratulations! Sounds like you are about to reinvent the multitaper method, in which a number of orthogonal window functions are used to get independent spectral estimates that are then averaged (with weights). A suitable window function family are the Slepian sequences: