Well, there’s not much to say about young Esther yet. I’ve not been able to find her family in “Buffalo, Texas.” Ancestry wants that to be in Coleman County, Wikipedia directs me to either Leon or Henderson County. There doesn’t appear to be a Buffalo county. And either way, no Ester Kate shows up. Lot’s of Kates though! One clue on the back of this photo says she is the sister of D. Smith, so we can make a slight assumption that Smith is her maiden name.
Only potential ID I found was a Kate or Kate M. born in 1871 to A. J. Smith a local judge in Rusk, Henderson Co., TX. She had a brother D. Smith (David or Duane depending on the census vs. Find A Grave.) Could be her but too early to tell. Good luck Esther Kate!
Esther Kate Smith
Unclaimed Ancestors 
There is a Buffalo, Texas. It is in Leon County. Population is 1,804. We travel through there from Brownsboro, Texas to get to I45 to travel to Houston, Texas.
Fairfield is nearby and also a possibility.
My mother, born in 1908, was an Esther Kate Smith also. Her parents were Andrew Jackson and Bertha Smith. She was born in Overton, Texas.