Motivated by the → anomalies, we study non-resonant searches for new physics at the large hadron collider (LHC) by considering final states with an energetic and hadronically decaying τ lepton, a b-jet and large missing transverse momentum (→ℎ+missT). Such searches can be useful to probe new physics contributions to →. They are analyzed not only within the dimension-six effective field theory (EFT) but also in explicit leptoquark (LQ) models with the LQ non-decoupled. The former is realized by taking a limit of large LQ mass in the latter. It is clarified that the LHC sensitivity is sensitive to the LQ mass for (1) TeV even in the search of →ℎ+missT. Although the LQ models provide a weaker sensitivity than the EFT limit, it is found that the non-resonant search of →ℎ+missT can improve the sensitivity by ≈ 40% versus a conventional mono-τ search (→ℎ+missT) in the whole LQ mass region. Consequently, it is expected that most of the parameter regions suggested by the →⎯ anomalies can be probed at the HL-LHC. Also, it is shown that R2 LQ scenario is accessible entirely once the LHC Run 2 data are analyzed. In addition, we discuss a charge selection of τh to further suppress the standard-model background, and investigate the angular correlations among b, τ and the missing transverse momentum to discriminate the LQ scenarios