Just FYI, the local area code and Jenny's number (867-5309) is a default that exists for most loyalty programs. So if you don't want to be tracked you can use that. I've heard that this was implemented for military folks who tend to be a lot more transient than regular locals.
Or just not use anything at all. Or do what I do, and prefer shopping at stores that don't have a loyalty program. In my area, stores that have loyalty programs also have higher prices so that the "discount" from the loyalty card makes the price roughly the same as at similar stores that don't have a loyalty program.
So I go to the stores that don't have one. That's not only better for privacy, but is more frictionless and comes with no price penalty.