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We use improved truncated Operator Product Expansion (OPE) for the Adlerfunction, involving two types of terms with dimension $D=6$, in thedouble-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules for the V+Achannel strangeless semihadronic $\tau$ decays. The generation of the higherorder perturbative QCD terms of the $D=0$ part of the Adler function is carriedout using a renormalon-motivated ansatz incorporating the leading UV renormalonand the first two leading IR renormalons. The trunacted $D=0$ part of the SumRules is evaluated by two variants of the fixed-order perturbation theory (FO),by Principal Value of the Borel resummation (PV), and by contour-improvedperturbation theory (CI). For the experimental V+A channel spectral function weuse the ALEPH $\tau$-decay data. We point out that the truncated FO and PVevaluation methods account correctly for the renormalon structure of the SumRules, while this is not the case for the truncated CI evaluation. We extractthe value of the ${\overline {\rm MS}}$ coupling $\alpha_s(m_{\tau}^2) =0.3235^{+0.0138}_{-0.0126}$ [$\alpha_s(M_Z^2)=0.1191 \pm 0.0016$] for theaverage of the two FO methods and the PV method, which we consider as our mainresult. If we included in the average also CI extraction, the value would be$\alpha_s(m_{\tau}^2) = 0.3299^{+0.0232}_{-0.0225}$[$\alpha_s(M_Z^2)=0.1199^{+0.0026}_{-0.0028}$]. This work is an extension andimprovement of our previous work [Eur.Phys.J.C81 (2021) 10, 930] where we usedfor the truncated OPE a more naive (and widely used) form and where theextracted values for $\alpha_s(M_Z^2)$ were somewhat lower.

Using improved Operator Product Expansion in Borel-Laplace Sum Rules with ALEPH $\tau$ decay data, and determination of pQCD coupling