there are others maybe who live in this area that may provide more info than what follows, but for what it's worth here are a couple of suggestions to get you started:
i have had some of the best hill country birding experiences ever at a place called chalk bluff park, just off hiway 55 (?) just outside of uvalde. it has a neat confluence of habitats, including upland s. texas scrub, a neat riparian corridor (a fine typical hill country stream runs along one boundry, and cedar/live oak orchard like forest. especially good in spring, when residents (lesser goldfinch, vermillion f/catcher, green k/fisher, common raven) migrant warblers (including golden cheek), vireos and flycatchers, and winter visitors mingle.
also, the water treatment plant has migrant shorebirds present, and garner state park is also nearby.