Let G = (V(G), E(G)) be a simple, finite and undirected graph of order n. Given a bijection f: V(G) ∪ E(G) → Zk such that for each edge uv ∈ E(G), f(u) + f(v) + f(uv) is constant C( mod k). Let nf(i) be the number of vertices and edges labeled by i under f. If |nf(i) - nf(j)| ≤ 1 for all 0 ≤ i < j ≤ k - 1, we say f is a k-totally magic cordial (k-TMC) labeling of G. A graph is said to be k-totally cordial magic if it admits a k-TMC labeling. In this paper, we give some ways to construct new families of k-TMC graphs from a known k-totally cordial magic graphs. We also give a sufficient condition for an odd graph to admit no k-TMC labeling. As a by-product, we determine the k-totally magic cordiality of many families of graphs.