  Let $X$ be a bundle over $S^1$ with fiber a 3--manifold $M$ and with
monodromy $\varphi$. Gay and Kirby showed that if $\varphi$ fixes a genus $g$
Heegaard splitting of $M$ then $X$ has a genus $6g+1$ trisection. Genus $3g+1$
trisections have been found in certain special cases, such as the case where
$\varphi$ is trivial, and it is known that trisections of genus lower than
$3g+1$ cannot exist in general. We generalize these results to prove that there
exists a trisection of genus $3g+1$ whenever $\varphi$ fixes a genus $g$
Heegaard surface of $M$. This means that $\varphi$ can be nontrivial, and can
preserve or switch the two handlebodies of the Heegaard splitting. We
additionally describe an algorithm to draw a diagram for such a trisection
given a Heegaard diagram for $M$ and a description of $\varphi$.
