The algorithm developed by the CMS Collaboration to reconstruct and identify $\tau$ leptons produced in proton-proton collisions at $\sqrt{s}=$ 7 and 8 TeV, via their decays to hadrons and a neutrino, has been significantly improved. The changes include a revised reconstruction of $\pi^0$ candidates, and improvements in multivariate discriminants to separate $\tau$ leptons from jets and electrons. The algorithm is extended to reconstruct $\tau$ leptons in highly Lorentz-boosted pair production, and in the high-level trigger. The performance of the algorithm is studied using proton-proton collisions recorded during 2016 at $\sqrt{s}=$ 13 TeV, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The performance is evaluated in terms of the efficiency for a genuine $\tau$ lepton to pass the identification criteria and of the probabilities for jets, electrons, and muons to be misidentified as $\tau$ leptons. The results are found to be very close to those expected from Monte Carlo simulation.