We consider a modified gravity framework for inflation by adding to the Einstein-Hilbert action a direct $f(phi)T$ term, where $phi$ is identified as the inflaton and $T$ is the trace of the energy-momentum tensor. The framework goes to Einstein gravity naturally when inflaton decays out. We investigate inflation dynamics in this $f(phi)T$ gravity (not to be confused with torsion-scalar coupled theories) on a general basis, and then apply it to three well-motivated inflationary models. We find that the predictions for the spectral tilt and the tensor-to-scalar ratio are sensitive to this new $f(phi)T$ term. This $f(phi)T$ gravity brings both chaotic and natural inflation into better agreement with data. For Starobinsky inflation, the coupling constant $alpha$ in $[-0.0026,0.0031]$ for $N=60$ is in Planck-allowed $2sigma$ region.
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