I think you have to be understanding about the fact that at this particular time in history it was a very shameful thing to have a child out of wedlock. I had always thought the stigma simply got worse and stronger as you went further back in time, but I don't think that was necessarily true, as I have found some very interesting comments on parish registers, which seemed quite accepting of children born out of wedlock, and yet others which were horrible.
Also, you always have to bear in mind that a child could have been the result of a rape. It isn't a nice thing to consider, but it could be that your grandma was protecting your mother from something unpleasant. In my family one of my grandmother's cousins didn't know that she was born as the result of rape, when I did. My grandmother had told me all about it; my cousin's grandmother had told her nothing about it. She never met her mother at all, and was brought up by her grandmother, with her grandmother's name.
Sometimes in family history you have to accept that a brick wall is a brick wall. It is possible that DNA testing may progress and tell us things we can't even dream about yet, so maybe it is "not yet" rather than never.
If it were me, I think I would be collecting as much as possible about the whole extended family - sisters and brothers, uncles and aunts, and trying to compile a time line to work out where granny was, how old the others in the family were, and what their movements were. Even so, it's a long shot to be able to gather enough circumstantial evidence to even make an educated guess.
I hope the rest of your family tree is very rich and rewarding to make up for the gap.