  We studied $\alpha$ cluster states in $^{26}$Mg via the
$^{22}$Ne($^{6}$Li,$d\gamma$)$^{26}$Mg reaction in inverse kinematics at an
energy of $7$ MeV/nucleon. States between $E_x$ = 4 - 12 MeV in $^{26}$Mg were
populated and relative $\alpha$ spectroscopic factors were determined. Some of
these states correspond to resonances in the Gamow window of the
$^{22}$Ne($\alpha$,n)$^{25}$Mg reaction, which is one of the main neutron
sources in the astrophysical $s$-process. We show that $\alpha$-cluster
strength of the states analyzed in this work have critical impact on s-process
abundances. Using our new $^{22}$Ne($\alpha$,n)$^{25}$Mg and
$^{22}$Ne($\alpha$,$\gamma$)$^{26}$Mg reaction rates, we performed new
s-process calculations for massive stars and Asymptotic Giant Branch stars and
compared the resulting yields with the yields obtained using other
$^{22}$Ne+$\alpha$ rates from the literature. We observe an impact on the
s-process abundances up to a factor of three for intermediate-mass AGB stars
and up to a factor of ten for massive stars. Additionally, states in $^{25}$Mg
at $E_x$ $<$ 5 MeV are identified via the $^{22}$Ne($^{6}$Li,$t$)$^{25}$Mg
reaction for the first time. We present the ($^6$Li, $t$) spectroscopic factors
of these states and note similarities to the $(d,p$) reaction in terms of
reaction selectivity.
