Let \(G\) be a graph with \(m\) edges and \(n\) vertices and \(f\) be an injection (called edge labeling) from the edge set of \(G\) to the set \(\{ 1,2,\dots ,m\}\). Let for any vertex \(x\) of \(G\) the weight of \(x\), \(w(x)\), be the sum of labels \(f(e)\) of all edges incident with \(x\). Then \(G\) is supermagic if there is a labeling \(f\) of the edges of \(G\) such that \(w(x)=\lambda \) for some positive integer \(\lambda\) and any vertex \(x\) of \(G\). \(G\) is called \((a,1)\)-antimagic if the labeling \(f\) has the property that the set of weights of all vertices of \(G\) is equal to \(\{a,a+1,\dots,a+n-1\}\) for some positive integer \(a\). It is shown that if \(G\) is an \((a,1)\)-antimagic graph decomposable into two edge-disjoint \(r\)-factors, then \(G\times K_2\) is supermagic. It is also shown that if \(F\) is an \((a,1)\)-antimagic \(r\)-regular graph, then under certain numerical conditions the join \(G\oplus K_1\) is supermagic. One more rather technical construction is also provided.