I didn't find anything in the resources I mentioned about your Mayme Smith.
The nearest I found was 2 children born to Albert & Lucinda Kingry Smith.
Since they are in the time frame you mentioned I'll include the info though I
see nothing to indicate a connection to your people. There are no given
names, but they had a girl born 28 Aug 1882 and a boy born 17 May 1884. They
were in Miami Co, but no township or town is noted.
Joyce