  Within the framework of six-dimensional ${\cal N}=(1,0)$ conformal
supergravity, we introduce new off-shell multiplets ${\cal O}{}^{*}(n)$, where
$n=3,4,\dots,$ and use them to construct higher-rank extensions of the linear
multiplet action. The ${\cal O}{}^{*}(n)$ multiplets may be viewed as being
dual to well-known superconformal ${\cal O}(n)$ multiplets. We provide
prepotential formulations for the ${\cal O}(n)$ and ${\cal O}{}^{*}(n)$
multiplets coupled to conformal supergravity. For every ${\cal O}{}^{*}(n)$
multiplet, we construct a higher derivative invariant which is superconformal
on arbitrary superconformally flat backgrounds. We also show how our results
can be used to construct new higher derivative actions in supergravity.
