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Within the framework of six-dimensional N=(1,0) conformal supergravity, we introduce new off-shell multiplets Oâ(n), where n=3,4,â¦, and use them to construct higher-rank extensions of the linear multiplet action. The Oâ(n) multiplets may be viewed as being dual to well-known superconformal O(n) multiplets. We provide prepotential formulations for the O(n) and Oâ(n) multiplets coupled to conformal supergravity. For every Oâ(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.