  We use improved truncated Operator Product Expansion (OPE) for the Adler
function, involving two types of terms with dimension $D=6$, in the
double-pinched Borel-Laplace Sum Rules and Finite Energy Sum Rules for the V+A
channel strangeless semihadronic $\tau$ decays. The generation of the higher
order perturbative QCD terms of the $D=0$ part of the Adler function is carried
out using a renormalon-motivated ansatz incorporating the leading UV renormalon
and the first two leading IR renormalons. The trunacted $D=0$ part of the Sum
Rules is evaluated by two variants of the fixed-order perturbation theory (FO),
by Principal Value of the Borel resummation (PV), and by contour-improved
perturbation theory (CI). For the experimental V+A channel spectral function we
use the ALEPH $\tau$-decay data. We point out that the truncated FO and PV
evaluation methods account correctly for the renormalon structure of the Sum
Rules, while this is not the case for the truncated CI evaluation. We extract
the 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 the
average of the two FO methods and the PV method, which we consider as our main
result. 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 and
improvement of our previous work [Eur.Phys.J.C81 (2021) 10, 930] where we used
for the truncated OPE a more naive (and widely used) form and where the
extracted values for $\alpha_s(M_Z^2)$ were somewhat lower.
