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The simplest $\alpha$-attractor $T$ model is given by the potential$V=V_0\tanh^2(\lambda\phi/M_{pl})$. However its generalization to the class ofmodels of the type $V = V_0 \tanh^p (\lambda \phi/M_{pl})$ is difficult tointerpret as a model of inflation for most values of $p$. Keeping the basicmodel, we propose a new generalization, where the final potential is of theform $V = V_0 (1-\sech^p (\lambda \phi/M_{pl}))$, which does not present any ofthe problems that plague the original generalization, allowing a successfulinterpretation as a model of inflation for any value of $p$ and, at the sametime, providing the potential with a region where reheating can occur for any$p$ (including odd and fractional values) without difficulty. In the cases $p =1, 2, 4$ we obtain the solutions $r(n_s, N_{ke})$ where $r$ is thetensor-to-scalar ratio, $n_s$ the spectral index and $N_{ke}$ the number of$e$-folds during inflation. We also show how these solutions connect to the$\phi^2$ monomial.

New generalization of the simplest $\alpha$-attractor $T$ model