The latest CMB data from ACT DR6, in combination with Planck, DESI, and BICEP/Keck, indicate a slight upward shift in the scalar spectral index. This trend puts several previously favored inflationary models under tension. In this work, we study an inflationary scenario in the framework of $f(R, T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor, with a nonminimal coupling between matter and curvature. The inflaton is assumed to be a noncanonical scalar field with a generalized kinetic energy. To analyze the dynamics of inflation, we employ the Hamilton-Jacobi formalism, where the Hubble parameter is expressed as a function of the scalar field rather than the potential. Within this setup, we examine two functional forms of the Hubble parameter, a power-law and an exponential form, and derive key observables such as the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$. Comparing the results with ACT DR6, we explore the parameter space of the model. We find that the power-law case fits the data well across a wide range of free parameters, while the exponential case requires a large number of e-folds to be consistent with observations. After inflation, we study reheating, where the dynamics of reheating and inflation are closely linked. Taking into account the overproduction of primordial gravitational waves constrained by the observational bound on $\Delta N_{\text{eff}}$, we obtain a lower limit on the reheating temperature, which is especially restrictive for the stiff equation of state $\omega_{\text{re}}$. This bound implies that the number of e-folds of inflation should generally not exceed $N \lesssim 64(65)$. The resulting energy spectrum of gravitational waves exhibits an enhanced amplitude, thereby bringing it within the observable range of upcoming detectors. We also check the consistency of the model with the Swampland conjectures and the Trans-Planckian Censorship Conjecture (TCC). Our results demonstrate that combining $f(R, T)$ gravity with noncanonical field dynamics provides a rich and testable framework for the early universe. In addition, the Hamilton-Jacobi approach, by avoiding extra approximations, yields a clearer picture of inflation in modified gravity and opens new directions for addressing fundamental problems in high-energy cosmology.
