A linearly ordered (LO) -colouring of a hypergraph assigns to each vertex a colour from the set {0, 1, . . . , −
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 -colouring of an LO 2-colourable 3-uniform hypergraph for any constant
≥ 2 [STACS’21] but even the case = 3 is still open. Nakajima and Zivn ˇ y gave polynomial-time algorithms for ´
finding, given an LO 2-colourable 3-uniform hypergraph, an LO colouring with ∗
(
√
) colours [ICALP’22]
and an LO colouring with ∗
(
3
√
) colours [ACM ToCT’23]. Very recently, Louis, Newman, and Ray gave an
SDP-based algorithm with ∗
(
5
√
) colours. We present two simple polynomial-time algorithms that find an
LO colouring with (log2
()) colours, which is an exponential improvement
