So we took Jess' car back to the shop and they said they'd look at it again without charging us anything until/unless they identified a problem.
They did a load test on the battery and, surprise, it wasn't providing enough current. (Well duh, it was dead and we had to jump start the car to get it to the shop.) So they put it on a charger for a few hours and then tested it again and it was fine. So they let it sit for several hours and then tested it again and it was still fine. So we had them keep it over the weekend and they tested it again Monday morning and it was still fine.
So we picked it up and came back home and it's been sitting since then until this afternoon when Jess needed to go somewhere again. And it started fine. So the problem seems to be fixed and the issue was just that the battery hadn't be charged up again after being dead originally.
Which just makes me wonder: When a customer brings in a car with a completely dead battery and you replace the alternator, shouldn't standard procedure be to hook the battery up to a charger to get it back to a full charge before having the customer pick up the vehicle?
Well, apparently not. So unless they did something else without owning up to it the residual problem was just an under-charged battery. Regardless, we didn't have to pay any more than we already had, so I at least don't have to be grumpy about that.
End of story.