We prove that the model checking problem for the existential fragment of first-order (FO) logic on partially ordered sets is fixed-parameter tractable (FPT) with respect to the formula and the width of a poset (the maximum size of an antichain). While there is a long line of research into FO model checking on graphs, the study of this problem on posets has been initiated just recently by Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon their result in two ways: (1) the runtime of our algorithm is O(f(|\phi|,w)*n2) on n-element posets of width w, compared to O(g(|\phi|)*n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow. We complement this result by showing that, under a certain complexity-theoretical assumption, the existential FO model checking problem does not have a polynomial kernel.