The accurate computation of a series of integer-order Bessel functions is often required in applications in engineering and physics. Numerous authors have shown how a recurrence relationship can be used in the backward and forward directions to accurately compute integer-order Bessel functions of the first and second kinds, respectively, for real- valued arguments. However, significant round-off errors can result when these standard recurrence algorithms are employed and the argument for the Bessel function is complex valued. C. F. du Toit recently developed an algorithm that overcomes these numerical instabilities for Bessel functions of the first and second kinds. However, if one needs to compute integer-order Hankel functions with complex arguments, then we have found that numerical round-off errors can lead to inaccurate results if Bessel functions of the first and second kinds are superimposed to obtain the desired Hankel functions. To address this problem, a technique is presented in this paper that uses a combination of forward and backward recurrence for the computation of integer-order Hankel functions with complex-valued arguments.