We study the electronic properties of a one-dimensional disordered chain made up of Pöschl-Teller potentials. The features of the whole spectrum of the random chain in the thermodynamic limit are analyzed in detail by making use of the functional equation formalism. The disordered system exhibits a fractal distribution of states within certain energy intervals and two types of resonances exist for the uncorrelated case. These extended states are characterized by different values of their critical exponents and different behaviors near the critical energies when the system is finite, but their existence can be defined by a single common condition. The chain is also considered including a natural model of short-range correlated disorder. The results show that the effects of the correlations considered are independent of the potential model.