That every (closed oriented) $3$-manifold has an open book decomposition is attributed to Alexander

Alexander, J. W. Note on Riemann spaces. American M. S. Bull. 26, 370-372 (1920).

and there are several explicit ways to construct open book decompositions on $3$-manifolds of which none will yield a simple open book in general.

I think most authors do not allow the binding to be empty in the definition of an open book.

For $S^1\times S^2$ there exists a preferred open book, namely the open book with trivial monodromy and annulus page.

For the other surfaces, one can see that the monodromy cannot be the identity and thus the open books have to be more complicated.

Pattrick Massot has beautiful animations of a simple open book on $T^3$ with $4$ binding components and a page of genus $1$:

https://www.math.u-psud.fr/~pmassot/en/exposition/giroux_correspondance.html

If you look at open books from another viewpoint there may be simpler open books. For example, it is known that on any $3$-manifold there exist open books with genus-$0$-pages or open books with only $1$ binding component.

I could imagine that one can construct open books on general $S^1\times \Sigma$ similar to that one above. Note, that in the open book of $S^1\times T^2$ we get one binding component for every handle of the surface $T^2$ in $S^1\times T^2$. If one starts with a genus-$g$ surface presented as a $2g$-gon with edges identified, one can probably draw a similar open book.