An adaptive distributed observer was recently proposed to solve the cooperative output regulation problem for discrete-time linear multi-agent systems. The validity of this approach relies on the assumptions that all the eigenvalues of the leader's system matrix are inside or on the unit circle and the solution to the regulator equations is unique. Besides, the range of the observer gains of the adaptive distributed observer is determined by the spectrum of a graph matrix of the communication network, which is not desirable in applications. To overcome these drawbacks, in this paper, we will first propose a new type of the discrete-time adaptive distributed observer for the leader system, whose observer gains are fixed and hence are independent of the communication network, and moreover, this observer allows the eigenvalues of the leader's system matrix to be inside a circle with radius strictly greater than one. Then, we will strengthen the existing adaptive control approach so that it can handle the case where the solution to the regulator equations is not unique. These improvements have not only made our approach more practical, but also enlarged the systems that can be handled by the existing approach.