I followed the suggestion in this piece and priced toilet paper and Kleenex at Jet.com. And I found what the author did: the prices at Jet were below Amazon's and substantially below the prices in my supermarket. I ordered enough to get free delivery, took advantage of the their limited offer of 15% off, and got exactly what I thought I'd ordered and on time. Saved a nice piece of change.
Other items beyond those two either weren't available or the prices didn't seem to be anything special. So I'm not switching from Amazon yet, but I've bookmarked Jet.com for further use.