Welcome to Alabama!
The Swan Creek Management Area near Decatur and Athens has a very nice outdoor range. IIRC, two 100 yd, one 50 and one 25. Someone who has been there more recently may correct me.
You need a hunting license and the mgt area permit to use the ranges, I believe. The guys at Larry's will probably know all the specifics, good people over there. Depending on which side of Huntsville you live on there are two easy routes to get there. I 565 towards Decatur to Hwy 31 or Hwy 72 through Athens and hang a left at the tractor dealership on the West side of town. If you need more detailed directions, let me know and I'll try to dust off the memory. Used to shoot there all the time, but haven't been there in years since I joined the range over here in the Shoals.
PS: If you do any competitive shooting, The North Alabama Shooting Association over here in Sheffield hosts four rimfire matches a month and one IDPA match a month. Matches are open to anyone and it's not too long of a drive. www.nasaclub.com for all the info.