In LHC searches for new and rare phenomena the top-associated channel pp→tt‾W±+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→tt‾W±+X that enter at O(αs3α) and O(αs4α), i.e., that contribute at NLO and NNLO in QCD in inclusive tt‾W± production. Using fixed-order computations, we estimate that a mixture of real and virtual corrections at O(αs4α) in well-defined regions of phase space can arguably increase the total tt‾W± 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 tt‾Z process.