A linearly ordered (LO) k-colouring of a hypergraph assigns to each vertex a colour from the set {0, 1, . . ., k − 1} in such a way that each hyperedge has a unique maximum element. Barto, Batistelli, and Berg conjectured that it is NP-hard to find an LO k-colouring of an LO 2-colourable 3-uniform hypergraph for any constant k ≥ 2 [STACS’21] but even the case k = 3 is still open. Nakajima and Živný gave polynomial-time algorithms for finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with O*(√n) colours [ICALP’22] and an LO colouring with O*(√3 n) colours [ACM ToCT’23]. Very recently, Louis, Newman, and Ray gave an SDP-based algorithm with O*(√5 n) colours. We present two simple polynomial-time algorithms that find an LO colouring with O(log2(n)) colours, which is an exponential improvement.