Construct Postfilter for overall Minimum or Linear Phase Filter
What you're trying to do is impossible to implement (i.e., if we require that the post-filter is causal and stable). If the original filter $h[n]$ is not already a minimum-phase filter, it has zeros outside the unit circle. These zeros need to be compensated for by the post-filter $g[n]$, i.e., the filter $g[n]$ must have poles outside the unit circle, and, consequently, it cannot be causal and stable.

On the other hand, constructing a linear phase filter using a post-filter is possible. Just make sure that for each zero $z_{0,i}$ of the filter $h[n]$, the post-filter $g[n]$ has a zero at $1/z^*_{0,i}$, i.e., for each zero you need another one mirrored across  the unit circle.