If I understand what you are asking for correctly I am assuming that your matrix S has the fft of the original time domain data.
I think you could then recover it and interpolate to get data for every second in the day using:
% make time vector for original spacing (300 samples per day)
t = linspace(0,86400-1,300);
% recover time domain signal
X = ifft(S);
% interpolate intermediate values
tq = 0:86400-1;
Xq = interp1(t,X,tq)
