We define and axiomatize three new logics based on the connexive logic $\mathsf{C}$, the modal logic $\mathsf{CnK}$ and the conditional logics $\mathsf{CnCK}$ and $\mathsf{CnCK}_R$. These logics display strong connexivity properties and are connected to one another, since $\mathsf{CnCK}_R$ is the reflexive extension of $\mathsf{CnCK}$ and $\mathsf{CnK}$ is faithfully embeddable into both $\mathsf{CnCK}$ and $\mathsf{CnCK}_R$ in a multitude of natural ways. We argue that all the three logics provide (albeit in different ways) natural expansions of $\mathsf{C}$ to their respective languages that preserve and further develop several core properties of $\mathsf{C}$, especially its connexivity profile.
