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A graph of order $n$ is distance magic if it admits a bijective labeling ofits vertices with integers from $1$ to $n$ such that each vertex has the samesum of the labels of its neighbors.  This paper contributes to the long term project of characterizing alltetravalent distance magic graphs. With the help of a computer we find that outof almost nine million connected tetravalent graphs up to order $16$ only nineare distance magic. In fact, besides the six well known wreath graphs there areonly three other examples, one of each of the orders $12$, $14$ and $16$. Weintroduce a generalization of wreath graphs, the so-called quasi wreath graphs,and classify all distance magic graphs among them. This way we obtaininfinitely many new tetravalent distance magic graphs. Moreover, the twonon-wreath graphs of orders $12$ and $14$ are quasi wreath graphs while the oneof order $16$ can be obtained from a quasi wreath graph of order $14$ using asimple construction due to Kov\'a\v{r}, Fron\v{c}ek and Kov\'a\v{r}ov\'a.

Tetravalent distance magic graphs of small order and an infinite family of examples