We develop a unified framework based on nonlinear sequence transformations to probe the asymptotic structure of perturbative QCD beyond fixed-order expansions. Focusing on the hadronic $\tau$-decay correction $\delta^{(0)}$, we employ complementary approaches—Levin-type transformations and Shanks transformations implemented via Wynn’s $\varepsilon$-algorithm and its generalizations, to resum the divergent FOPT series and extract higher-order information. These methods generate rational approximants that encode the large-order behavior of the series, yielding stable and consistent predictions for the coefficients $c_{n,1}$ across independent transformation schemes. We show that Levin-type transformations, through explicit remainder estimates, and Shanks-type transformations, via elimination of leading asymptotic errors, provide complementary reconstructions of the same underlying structure. Together, they capture renormalon-driven factorial growth and provide a data-driven approximation to the underlying Borel behavior without explicit multi-loop input. The resulting resummed predictions for $\delta^{(0)}$ exhibit improved convergence and reduced truncation uncertainties. Our results establish sequence transformations as a systematic, model-independent bridge between finite-order perturbation theory and its asymptotic completion, with direct implications for precision QCD phenomenology.