A $k$-uniform tight cycle is a $k$-uniform hypergraph with a cyclic ordering of its vertices such that its edges are all the sets of size $k$ formed by $k$ consecutive vertices in the ordering. We prove that every red-blue edge-coloured $K_n^{(4)}$ contains a red and a blue tight cycle that are vertex-disjoint and together cover $n-o(n)$ vertices. Moreover, we prove that every red-blue edge-coloured $K_n^{(5)}$ contains four monochromatic tight cycles that are vertex-disjoint and together cover $n-o(n)$ vertices.