Sometimes the floor mats are held in place with the same screws which hold bits of plastic trim in place. Even if there aren't actual screw holes you may need to loosen bits of trim to get them back easily.
If this is more trouble than it seems worth you could instead carefully cut off the flap and use something like double sided tape, self adhesive Velcro or contact adhesive to hold it in place.
I used to work for a van conversion company and often had to remove mats and trim to fit various bits and pieces. Often trim fixings are designed for one time easy assembly (often using plastic clips etc) and don't go back together perfectly once removed.
A common problem is that to remove one b it of trim you end up with a complex jigsaw puzzle of other bits which need to be removed first.
I ended up accumulation a kit of various plastic wedges, prying tools, hooks etc for problematic trim.