The ETH model of the pion-nucleon ($\pi N$) interaction contains $t$-channel $\sigma$- and $\rho$-exchange graphs, as well as the $s$- and $u$-channel contributions with the well-established $s$ and $p$ baryon states with masses below $2$ GeV as virtual particles; the model amplitudes obey crossing symmetry and isospin invariance. In the present work, we give the analytical expressions for the model contributions to the $K$-matrix elements up to (and including) the $f$ waves. We also extract a new phase-shift solution after performing a partial-wave analysis of meson-factory $\pi^\pm p$ elastic-scattering data below $100$ MeV; included in our results now are also the effects of the variation of the $\sigma$-meson mass in the interval which is currently recommended by the Particle-Data Group. Finally, we revisit the subject of the $\pi N$ $\Sigma$ term and, using the model amplitudes, obtain $\Sigma = 72.4 \pm 3.1$ MeV. Our prediction agrees well with the result extracted with Olsson's method, after a few flaws in his paper [Phys. Lett. B 482 (2000) 50] were corrected.