Amy wrote:

For the past few months, it seems that periodically, one of my
cats would urinate right outside of the litter box. I managed to
catch this happening once - she would go inside the litter box,
and stand right by the edge and urinate, resulting in puddles
right outside of the box.

Currently I have two cats. Underneath and in the surrounding area
is a vinyl carpet protector. When I remove stuff from the box
during the day, the tracking on the vinyl carpet protector is
emptied into the box. Occasionally the carpet protector is
saturated with a cleaner and then dried with recycled paper towels
(usually just used for soaking up water around the sink). At the
same time, the litter is allowed to diminish in the box, then the
same cleaner is applied to the sides and bottom edges, again dried
with paper towels. That done to all four sides. Litter dust tends
to run up the sides and over the edges. That may or may not be
your problem, but the idea that a dirty area makes for less
accurate urination/defecation makes sense to me.

Good luck.