The Ising model on fractal lattices has attracted renewed interest, yet a quantitative understanding of how the critical temperature depends on the Hausdorff dimension $\left(D_{\mathrm{H}}\right)$ and connectivity $\left(Q\right)$ remains limited. We study the ferromagnetic Ising model on the one-parameter “legs-extension” family of fractal lattices originally introduced by Genzor et al. and apply the higher-order tensor renormalization group (HOTRG) method, following the tensor-network construction developed for these geometries in earlier work. By tuning a single structural parameter, we access lattices with $D_{\mathrm{H}}$ decreasing from $1.792$ to $1.260$ and connectivity $Q$ from $0.50$ to $0.131$, and obtain high-precision critical temperatures $T_{\mathrm{c}}$ and magnetization exponents $\beta$ with carefully quantified uncertainties from systematic bond-dimension and fitting-window analyses, finding that $\beta$ is strongly and almost exponentially suppressed as $D_{\mathrm{H}}$ is reduced. Along this one-dimensional trajectory in the $\left(D_{\mathrm{H}},Q\right)$ plane, $T_{\mathrm{c}}$ decreases monotonically with $D_{\mathrm{H}}$ and, when plotted against $Q$, exhibits a robust near-linear behavior within this lattice family over almost an order of magnitude in connectivity. A finite-temperature phase transition persists throughout the explored range down to our lowest Hausdorff dimension, $D_{\mathrm{H}}\approx 1.260$, implying that the lower critical dimension of this family, if it exists, lies below this value. Comparison with zero-order and first-order mean-field theory shows that HOTRG systematically incorporates fluctuation effects beyond mean-field approximations. The tabulated $T_{\mathrm{c}}$ and $\beta$ values thus provide a stringent benchmark for theories of criticality on fractal lattices and a quantitative reference for future numerical and analytical studies on other complex geometries.

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• Fractals
• Ising model
• Tensor network renormalization