Didn't want to wait for Steam to update. So figured out a process to install the patch over Steam (which was at version 1.1.0.7).
1) In your Steam Library, go to the properties for DCS A-10. Under the Updates tab, change it to "Do not automaticially update this game".
2) Find your folder for A-10.
For example: My folder name:
C:\Program Files (x86)\Steam\steamapps\common\dcs a-10c warthog
3) Launch patch_setup_v1.1.0.8.exe
4) For the path to your game, use the folder browser GUI to navigate to the dcs A-10 warthog directory where your game is installed.
5) When you click ok from the folder browser, double-check to ensure it hasn't added a "New Folder" folder to the path. For example, when I clicked ok, the directory listed for the new install was:
C:\Program Files (x86)\Steam\steamapps\common\dcs a-10c warthog\New Folder
Remove the New Folder so the path is back to:
C:\Program Files (x86)\Steam\steamapps\common\dcs a-10c warthog
6) When you click ok, you'll get a prompt "The Folder:
<Your folder path>
already exists. Would you like to install to that folder anyway?" - Click yes
7) When the patch completes - launch the game as you normally would
