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Within the framework of six-dimensional N = (1, 0) conformal supergravity, we introduce new off-shell multiplets O*(n), where n = 3, 4,..., and use them to construct higher-rank extensions of the linear multiplet action. The O*(n) multiplets may be viewed as being dual to well-known superconformal O(n) multiplets. We provide prepotential formulations for the O(n) and O*(n) multiplets coupled to conformal supergravity. For every 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. (C) 2017 The Author(s). Published by Elsevier B.V.