Why estimate linear and full (linear, quadratic, and correlational) selection coefficients separately?
This is directly following the advice of Lande &amp; Arnold (1983), saying: 


  Linear multiple
  regression can be used first to estimate the forces of directional selection, $\beta$, and their standard errors. Then a quadratic multiple regression (16) or (Al), can be used to estimate the forces of stabilizing selection, $\gamma$, with their standard errors. The regression (16) provides the best quadratic approximation to the selective surface (although valid estimates of $\beta$ can be obtained
  only from a purely linear regression, or by using the orthogonal regression (Al)).


The reason is that the estimates of direction selection is influenced by the higher order terms in the full model that includes quadratic and correlational selection, and the linear model therefore has the best estimates of the change in mean value over a generation of selection ($s = \bar{z}_{after} - \bar{z}_{before}$). However, the full model is the better representation of the fitness surface. There are also other methods to approximate the fitness surface.

For the record, I used the same approach in my master's thesis many years ago.