I've hunted goats in SD a bunch...pick your times wisely...Mid Sept to rifle opener is an absolute circus in some parts with all the guys running around chasing goats. If you don't mind walking get away from roads in big sections and you will get at unpressed animals. It's a good time I'd recommend it.
A lot of counties without much public have a good number of landowners that will let you hunt archery antelope. A friendly handshake and a handle of crown goes a long ways.