We report a search for single top quark production with the CDF II detector using 2.1  fb−1 of integrated luminosity of pp collisions at s√=1.96  TeV. The data selected consist of events characterized by large energy imbalance in the transverse plane and hadronic jets, and no identified electrons and muons, so the sample is enriched in W→τν decays. In order to suppress backgrounds, additional kinematic and topological requirements are imposed through a neural network, and at least one of the jets must be identified as a b quark jet. We measure an excess of signal-like events in agreement with the standard model prediction, but inconsistent with a model without single top quark production by 2.1 standard deviations (σ), with a median expected sensitivity of 1.4σ. Assuming a top quark mass of 175  GeV/c2 and ascribing the excess to single top quark production, the cross section is measured to be 4.9+2.5−2.2(stat+syst)  pb, consistent with measurements performed in independent data sets and with the standard model prediction.