On the San Saba/Mills county line. Google driving directions to Regency will get you there; it's closer to pavement from the FM500 (San Saba County) side, but the historical marker is at the Mills county end of the dirt road.
You're pretty much limited to the bridge itself and the roadway, but there might be some property owners willing to allow shoots from the river banks with the bridge in the background. It's a dirt, county road, and we only saw one other vehicle in the half-hour or so that it took to drive to the bridge, look it over, get a couple of pics, and drive out, so traffic shouldn't be a big issue.
The Swinging Bridge Camp is on the north bank, and should have a good view of the bridge, though at a high angle. Their number is 325-938-5670 if anyone is interested in checking on the possibility of a shoot from there.
Also in the area, about 15 miles away, just north of San Saba on a paved county road is the Beveridge Suspension Bridge which, while much smaller, is open only to foot traffic, so you can safely set up whatever is needed.
In retrospect, this could be a good place for an engagement session or outdoor formals if anyone has clients in the area. The closest towns are Brady, San Saba, and Goldthwaite, none of which are exactly large cities, though, so demand is probably pretty limited.
Neither is a place I'd plan a special trip to unless I just desperately needed a suspension bridge for a shot, but if you're going to be passing through the area, they're worth a side trip. Map This links on the gallery pages give more specific locations.