Summary: We show that, for each chordal graph \(G\), there is a tree \(T\) such that \(T\) is a spanning tree of the square \(G^2\) of \(G\) and, for every two vertices, the distance between them in \(T\) is not larger than the distance in \(G\) plus 2. Moreover, we prove that, if \(G\) is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree \(T\) of \(G\) that is an additive 3-spanner as well as a multiplicative 4-spanner of \(G\). In all cases the tree \(T\) can be computed in linear time.