If you make a private tree, you might get leaf hints on it, but you probably won't be able to form DNA circles. I think your best bet is to make a public tree, but mark it as a "test tree" or "do not use" tree so you warn others from using your tree as a source for their own research.
Then find your closest match with a healthy tree. If they are third cousins, look at their 3rd great grandparents. Chances are one of these sets is also your 3rd great grandparents and your mother's great, great grandparents. To speed things up, you might check where these grandparents were born/lived/died to see if it's anywhere in the area where your mother was born. I'd start with the set who physically lived closest to your mother's birthplace.
Create a tree with this set of ggg grandparents, their parents, and their grandparents. and attach this tree to you DNA profile. (I think you can put yourself, your mom, and "unknown grandparent" and "unknown great grandparent" etc, up to attach to them.) If you do this, I think you will get a leaf hint from the cousin whose tree you borrowed (and it will therefore not be "reliable"), but if you get more leaf hints it will mean there are others sharing your DNA, and they also have these ancestors in their tree. Plus you can then check and see who the common ancestor is. If you don't get other hints, you can either add another generation back, or try a different branch from your closest cousin's tree.
It will most likely be a time-consuming process, but I don't know of a better way to build a tree under the given circumstances.