It was recently observed in [1], that in index coding, learning the coding matrix used by the server can pose privacy concerns: curious clients can extract information about the requests and side information of other clients. One approach to mitigate such concerns is the use of $k$-limited-access schemes [1], that restrict each client to learn only part of the index coding matrix, and in particular, at most $k$ rows. These schemes transform a linear index coding matrix of rank $T$ to an alternate one, such that each client needs to learn at most $k$ of the coding matrix rows to decode its requested message. This paper analyzes $k$-limited-access schemes. First, a worst-case scenario, where the total number of clients $n$ is $2^T-1$ is studied. For this case, a novel construction of the coding matrix is provided and shown to be order-optimal in the number of transmissions. Then, the case of a general $n$ is considered and two different schemes are designed and analytically and numerically assessed in their performance. It is shown that these schemes perform better than the one designed for the case $n=2^T-1$.