Unlike output-controllability of linear systems which is a consequence of controllability, the output-controllability problems of nonlinear systems can be much more difficult than the controllability problems. Thus, it is a challenging task to derive criteria, especially algebraically verifiable ones, for output-controllability of nonlinear systems. In this paper, we consider output-controllability of a class of fundamental driftless discrete-time bilinear systems. We show that output-controllability of such systems cannot be fully deduced by controllability; however, near-controllability plays a key role in studying the output-controllability problem. A general necessary and sufficient condition as well as an algebraically verifiable sufficient condition is obtained for output-controllability of the systems via near-controllability. Our study reveals that output-controllability of a nonlinear system needs to be determined by its near-controllability instead of its controllability. Furthermore, inspired by near-controllability, we introduce a new notion, called output-near-controllability, for nonlinear systems and demonstrate it on the bilinear systems. Necessary as well as sufficient algebraic conditions for output-near-controllability of the bilinear systems are presented. Examples are given to demonstrate the conceptions and results of this paper.