Recently, Jafari and Sehgal obtained an extension of the Himmelberg fixed point theorem based on the Kakutani fixed-point theorem. We give generalizations of the extension to almost convex sets instead of convex sets. We also give generalizations for a large class 𝔅 of better admissible multimaps instead of the Kakutani maps. Our arguments are based on the KKM principle and some of previous results due to the second author.