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For a curve, that's all of course, since the polynomial is linear. Now let's say $X$ is a smooth surface with ample divisor $H$, we have the Hilbert polynomial $$\chi(\mathcal{O}_X(nH))= \frac{1}{2}nH(nH-K) + \chi(\mathcal{O}_X)$$ by Riemann-Roch. So the linear coefficient gives you the degree of the canonical divisor. In higher dimensions, the more general form of Riemann-Roch $$\chi(\mathcal{O}_X(nH)) = \int_X ch(\mathcal{O}(nH))td(X)$$ tells you that you're basically getting certain Chern numbers in $X$ and $H$ as coefficients.