I made some stay-dry liners the other day for my flats and diaper covers. I used some microfleece baby blankets that worked pretty well. I cut it up equally about 14" x 7 1/2" and the other strips I cut in half so it's 7" x 7 1/2 and sewn those two together.
But instead of creating some sort of "sock" to insert the folded flats, I kept both ends open to make removing them easier when I dump them in the washer.
flats padfolded and inserted inside the microfleece liner with a Kawaii Diaper Cover.
Trimness of a Kawaii diaper cover with a folded flat.