An answer similar to that of Simon Henry. Take (left) $M$-sets for a monoid $M$: the terminal is a singleton set, so has only two subobjects, and the only nontrivial propositional in your sense modality you can get is double negation. However there can be other modalities: $\Omega$ can be taken to be the Heyting algebra of left ideals of $M$, i. e. $M$-subsets $\mathfrak a\subseteq M$ of $M$ with the action on itself by left multiplication. This $\Omega$ is an $M$-set via
$$
m\mathfrak a=\{m'\in M\mid m'm\in\mathfrak a\}
$$
and it is clearly an $M$-equivariant Heyting algebra, i. e. $M$ acts on it via Heyting algebra endomorphisms. Thus for example analogs of all kinds of propositional modalities that you list can be realized in it, e. g. for an ideal $\mathfrak a\in\Omega$ we have modalities $\mathfrak a\cup-$, $\mathfrak a\Rightarrow -$, $(-\Rightarrow\mathfrak a)\Rightarrow\mathfrak a$ which in general cannot be obtained from any subobjects of the terminal, i. e. they are essentially "$M$-propositional" rather than "$1$-propositional".

**Caveat** - Simon Henry explains in the comment below why the specific examples I propose in the end are actually wrong. Currently trying to come up with some correct examples.