For a graph G, its k-rainbow independent domination number, written as γrik(G), is defined as the cardinality of a minimum set consisting of k vertex-disjoint independent sets V1,V2,…,Vk such that every vertex in V0=V(G)\(∪i=1kVi) has a neighbor in Vi for all i∈{1,2,…,k}. This domination invariant was proposed by Kraner Šumenjak, Rall and Tepeh (in Applied Mathematics and Computation 333(15), 2018: 353–361), which aims to compute the independent domination number of G□Kk (the generalized prism) via studying the problem of integer labeling on G. They proved a Nordhaus–Gaddum-type theorem: 5≤γri2(G)+γri2(G¯)≤n+3 for any n-order graph G with n≥3, in which G¯ denotes the complement of G. This work improves their result and shows that if G≇C5, then 5≤γri2(G)+γri2(G¯)≤n+2.