There are many reasons why you would prefer your first-shell fit to have your fit include high-k data, when feasible. I’ll name one, just to give you the idea:

Suppose you have a substance where you have a good estimate of S02 (transferability from a similar substance measured under the same conditions), but you don’t know the first-shell coordination number or sigma^2. Coordination number and sigma^2 are correlated, since they both primarily affect the amplitude of chi(k), but they are not completely correlated, as the coordination number has the same effect across the entire k-range, wherease sigma^2 has a much greater effect at greater k (in the EXAFS equation, it’s weighted by k^2). So extending the fit up to higher k reduces the correlation between coordination number and sigma^2, which can be very important to understanding the structure of your material.

Of course, there are other ways to reduce that correlation, such as by measuring a series of spectra at different temperatures. But it does provide a good example of why high-k information is useful when you can get it.

Best,