The top card is from the 1930s, C.T. American Art Colored (that’s just a phase Curt Teich Post Cards was going through in the 30s), published by Asheville Post Card Co., Asheville, N.C.
This is the bathhouse/swimming pool. The big Tate Springs Hotel is behind the photographer. Look closely, there are quite a few people enjoying a sunny day. On the back (undated) is “Dear Mrs. Nance Please accept my thanks to you and the entire class for the beautiful flowers and the ice cream. I hope to be with you soon. Much Love, Louise” Address is Mrs. A.M. Nance, Rutledge, Tenn. The numbers are in a different hand (the pencil lead is chipped, too). The one cent stamp places it in the right decade, too.
The bottom card, certainly just a decade younger, shows no people (they could have easily been airbrushed out), but all else seems to be the same. It’s linen-finished, also published by Asheville Post Card Company and I thought it was also printed by Curt Teich, but the inventory number doesn’t fit. This one also refers to Kingswood as being “near Morristown”, while the top one refers to “Tate Springs, Tenn.”