We present a search for a new heavy particle X produced in association with a top quark, pp[over ¯]→t(X→t[over ¯]q) or pp[over ¯]→t[over ¯](X[over ¯]→tq[over ¯]), where q stands for up quarks and down quarks. Such a particle may explain the recent anomalous measurements of top-quark forward-backward asymmetry. If the light-flavor quark (q) is reconstructed as a jet (j), this gives a t[over ¯]+j or t+j resonance in tt[over ¯]+jet events, a previously unexplored experimental signature. In a sample of events with exactly one lepton, missing transverse momentum and at least five jets, corresponding to an integrated luminosity of 8.7 fb(-1) collected by the CDF II detector, we find the data to be consistent with the standard model. We set cross-section upper limits on the production (pp[over ¯]→Xt or X[over ¯]t[over ¯]) at 95% confidence level from 0.61 pb to 0.02 pb for X masses ranging from 200 GeV/c(2) to 800 GeV/c(2), respectively.