Hi,
50 cM is on the lower end for a 3rd Cousin DNA match, only 17% probability, check the DNA painter shared cM tool for the full list of possible relationships. So it may be possible to solve your mystery, but it may take some work.
Based on you saying you hired a DNA expert, i am assuming you are not very experienced in using your DNA results to find relationships ? (apologies if that's not correct). The trick is to not think of DNA as a magic bullet that will immediately solve your question, but to think of it as way to get some very good clues. My suggestion will be to build out a tree for the match, and then look at shared matches you both share and try to fit into that tree. By doing this you can hopefully narrow down the ancestors that you may have in common. With this you can then start to work out the possible connection points to your line. So for example if your match has shared matches, which result in a common ancestor for them around the born around 1850's, you would know that the chances are one of the children for that common ancestor is likely to be the parent of your GGM.
Richard