Summary: A \((p,q)\) graph \(G\) is said to be magic if there exists a bijection \(f:V\cup E\to\{1,2, \dots, p+q\}\) such that for all edges \(xy\), \(f(x)+f(y) +f(xy)\) is a constant. Such a bijection is called a magic labeling of \(G\). The magic strength of a graph \(G\) is denoted by \(m(G)\) and is defined as the minimum of all constants where the minimum is taken over all magic labelings of \(G\). In this paper, we introduce almost-magic labeling, relaxed-magic labeling, almost-magic strength and relaxed-magic strength of a graph. We establish the magic strength of Huffman trees HT\(_n\), twigs TW\(_n\) \((n\) is odd) and the almost-magic strength of \(nP_2\) \((n\) is even) and twigs TW\(_n\) \((n\) is even). Also, we obtain bounds for magic strength of the path-union \(P_n(m)\) and the relaxed-magic strength of \(kS_n\) and \(kP_n\).