I have 2 people suggested as 3rd cousins by ancestry. They share 153 cM across 7 segments and 132 cM across 7 segments.
They do not match each other and me at 4 to 6 cousin level, maybe they do at 5 to 8 cousin level. However, they are both actual 3rd cousins, descended from our mutual 2G grandparents.
DNA inheritance is variable, but 140 cM seems quite a lot. How much do you share with the 2nd cousin match? Considerably more, or just a little bit to get her over the threshold between 3rd and 2nd cousin? They could be of the same generation or a different generation, and I think it will help matters if you can verify this.
http://thednageek.com/the-limits-of-predicting-relationships-using-dna/I think you may have hit the nail on the head about your great grandfather, especially since he was first born son, and born so soon after the marriage of your 2G grandmother. And in a place that you can relate to. Plenty of marriages in my family and a baby born a few months later, but usually as soon as the pregnancy would have been found out about. A marriage a few days before the birth does indicate the possibility of a different father.
Now you have found a possible name for your ancestor, I would follow him and build a tree assuming that it is him, whilst not as yet connectIng him to your tree. That way, if anyone does reply to your messages, you will have an idea who, what and where you are talking about.
I have done that with a family in my tree. My 2G grandfather John Steer 1814 gave his father's name as George Dawson when he married. His brother 1816 gave father's name as Dawes Steer. I have found a George Dawes in the same village in census, married 1816. Whilst realising this may be a complete red herring I have followed George Dawes and all his children for as many generations as possible, hoping for a DNA match. No luck so far, but you already have the DNA match.
Good luck,
Regards Margaret