Let $f: X \to C$ be a smooth surjective morphism from a smooth projective complex variety onto a smooth curve, and let $F$ denote a general fiber. I would like to show that $$\kappa(X) \leq \kappa{(F)} + 1$$ where $\kappa(X)$ is the Kodaira dimension. (Say, defined to be the largest dimension of the image of $X$ under the rational maps given by the linear systems $|mK_X|$ for $m \geq 1.$) We also assume that $C$ has genus $g(C) \geq 2$ and that a general fiber has $H^0(F,\mathcal{O}_F(K_F)) \neq 0$(ie. positive geometric genus).
For context, this claim appears in Lazarsfeld's Positivity II on page 51, example 6.3.60. The last assumptions listed above are used in the proof of the opposite inequality, so they may not be necessary for this direction since Lazarsfeld remarks the above inequality holds in more generality.
I'm having trouble figuring this one out. Since $C$ is a curve, it seemed natural to treat $F$ as a divisor where we have the adjunction formula $K_F = K_X + F|_F$. This however doesn't immediately work, since comparing global sections would require understanding/bounding $h^1(X, mK_X)$ and it's not clear to me how we can do this, since $K_X$ is not to my knowledge assumed or known to be positive in any way.
Perhaps there is some other way to compare the Kodaira map of $X$ and $F$, but it's not clear to me, especially since the Kodaira dimension can both increase or decrease upon restriction. (Say, a canonical curve in $\mathbb{P}^2$, or the exceptional curve of the blowup of a general type surface.)
Any help is appreciated!