I have to find the surfaces (up to isomorphism) that are connected 3-fold covers of $X=T^2 \setminus D^2$. Using the Euler characteristic criterion, I can see that the Euler characteristic of a possible cover must be $-3$. Based on this information, I can conclude that $M_{1,3}$ -- the orientable surface of genus $1$ with $3$ boundary components (torus minus $3$ disks) -- is a cover, which is an expected one just by drawing pictures also. But I cannot determine/rule out the other surfaces that arise using the Euler characteristic condition alone.
Any hints/suggestions would be greatly appreciated. Thanks!
$\textbf{Edit:}$ I want to add that I'm inclined to think that the only possible covering as required above is $M_{1,3}$. I was also thinking about this problem by "fattening" the $3$-fold covers of $S^1 \vee S^1$, which makes me say so. But I cannot prove this very formally.
Let me alter the problem ever so slightly by defining $X = T^2 \setminus \text{interior}(D^2)$, which makes $X$ a compact surface-with-boundary, of genus $1$ and with $1$ boundary component. And thus, as you say, $\chi(X)=-1$. And then, as you say, for any connected 3-fold covering map $M \mapsto X$ we must have $\chi(M)=-3$.
But the other thing you have to do is to consider the restricted 3-fold covering map $\partial M \mapsto \partial X$. There are several cases to consider, based on the number of components of $\partial M$, which can be $3$, $2$ or $1$.
Case 3. $\partial M$ has 3 components, each covering $\partial X$ with degree $1$. In this case $M$ has genus $1$, i.e. $M$ is the torus minus the interiors of three disjoint discs. This is the case you found.
Case 2. $\partial M$ has 2 components, one covering $\partial X$ with degree 2, and the other covering $\partial X$ with degree 1. This is not possible for an oriented surface of Euler characteristic equal to $-3$.
Case 1. $\partial M$ has 1 component, covering $\partial X$ with degree 3. In this case $M$ has genus $2$, i.e. $M$ is the closed surface of genus $2$ minus the interior of a single disc.
In summary, Case 2 does not occur, but cases 1 and 3 certainly occur, as one can show by direct construction of appropriate covering maps.