Shamir's Secret Sharing works by sharing data points on a curve, whereby when you have the required number of data points, you can find the function of the curve and find out the secret, which is effectively f(0).
In the polynomial form of the function, the constant (the coefficient of $x^0$) is the secret information.
Could you treat the coefficients of the other powers of $x$ as secret values as well?
If so, are they just as secure / secret? Or is there some subtlety going on where you need fewer values to get the other coefficients?
It seems that if you had a cubic curve that you could have 4 secrets $S_i$, instead of 3 random numbers and one secret:
$$f(x) = S_1x^3 + S_2x^2+S_3x+S_4$$