We present a progress on local computation algorithms for two coloring of $k$-uniform hypergraphs. We focus on instances that satisfy strengthened assumption of Local Lemma of the form $2^{1-αk} (Δ+1) e < 1$, where $Δ$ is the bound on the maximum edge degree of the hypergraph. We discuss how previous works on the subject can be used to obtain an algorithm that works in polylogarithmic time per query for $α$ up to about $0.139$. Then, we present a procedure that, within similar bounds on running time, solves wider range of instances by allowing $α$ at most about $0.227$.