This site does not allow advertising, but you might try
http://www.wingsinflight.com/birdres.html. They have links to lodging for birders, but it is not specific to the Hill Country. The person who runs the site lives in North Texas, and there are a lot of links to the Lower Rio Grande Valley. But that site might give you some good answers to your question.
Welcome to TBC. Be sure to post some photos of the birds (and animals) you see around your place.