  A new approximation scheme to the centrifugal term is proposed to obtain the
$l\neq 0$ bound-state solutions of the Schr\"{o}dinger equation for an
exponential-type potential in the framework of the hypergeometric method. The
corresponding normalized wave functions are also found in terms of the Jacobi
polynomials. To show the accuracy of the new proposed approximation scheme, we
calculate the energy eigenvalues numerically for arbitrary quantum numbers $n$
and $l$ with two different values of the potential parameter
$\sigma_{\text{0}}.$ Our numerical results are of high accuracy like the other
numerical results obtained by using program based on a numerical integration
procedure for short-range and long-range potentials. The energy bound-state
solutions for the s-wave ($l=0$) and $\sigma_{0}=1$ cases are given.
