You're describing my Great Grandma, Sophie Tucker.
She always used Dumolt as her maiden name, not Rhein, though a listing for Lizzy Rhein is in the Tucker Family bible. (I wondered who that was. Great Grandma Tucker died in 1952 and the connection wasn't clear)
Great Grandma Tucker only had three kids: Louise, Merlin & Urban. I'm descended from Louise, thru her only son John. So your Mom must be one of my Dad's cousins. But I digress . . .
As I understood it, Joseph Dumolt was the man that married her mom after the affair & it's fall out. But this didn't happen in 1865. Sophie married Great Grandpa Tucker in 1904 or 05, she was slightly under the legal age, & was widowed relatively young, in 1929.
She had a lot of resentment towards her step-dad and was unhappy @ home. So she ran off w/John and moved to Central Illinois, then Indianapolis where she was widowed and finally Southern California where she passed away.
The facts you give jibe except for the general time frame. Now, my family might be wrong about Joseph Dumolt. She kind of kept him in the shadows and I've been curious about him since childhood.
I'm anxious to hear what you know. I hope you see this & email me soon.