In LHC searches for new and rare phenomena the top-associated channel $pp to toverline{t}W^pm +X$ is a challenging background that multilepton analyses must overcome. Motivated by sustained measurements of enhanced rates of same-sign and multi-lepton final states, we reexamine the importance of higher jet multiplicities in $pp to toverline{t}W^pm +X$ that enter at $mathcal{O}(alpha_s^3alpha)$ and $mathcal{O}(alpha_s^4alpha)$, i.e., that contribute at NLO and NNLO in QCD in inclusive $toverline{t}W^pm$ production. Using fixed-order computations, we estimate that a mixture of real and virtual corrections at $mathcal{O}(alpha_s^4alpha)$ in well-defined regions of phase space can arguably increase the total $toverline{t}W^pm$ rate at NLO by at least $10%-14%$. However, by using non-unitary NLO multi-jet matching, we estimate that these same corrections are at most $10%-12%$, and at the same time exhibit the enhanced jet multiplicities that are slightly favored by data. This seeming incongruity suggests a need for the full NNLO result. We comment on implications for the $toverline{t}Z$ process.
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