As you've probably discovered, there are LOTS of Pottingers in this area. I have them in my tree and it's a bit of a minefield!
The Berkshire Burial Index has a burial for a Hannah Pottinger in Chieveley 28.7.1791. Helpfully

it describes her as the wife of [blank] but at least you know she was somebody's wife! (not an infant). So, possibly 'your' Hannah.
As well as the one you've already got in Compton, there are these burials around the right time (1796-1805) for a John Pottinger:
22 Aug 1796 Blewbury John Pottinger of Upton
29 Sep 1799 Bradfield John Pottinger (no additional information)
Bradfield is less than 10 miles from Chieveley, and East Ilsley is on the third point of a 10 mile triangle with the other two.
I don't know how much you know about these Pottingers, but, in my experience, the Compton Pottingers tend to have been very well to do.
Jill