Double sided tape will work. Actually quite well but is expensive and difficult to remove.
Another option is a layer of normal packing tape applied in strips to the waste board & underside of part, then superglue between the two layers of tape. Holds surprisingly well due to the large surface area and then can be pulled off easily enough as its only standard packaging tape touching the parts. The super glue stays between the two layers of tape and is does not need to be pulled apart (which would not happen!)
Clamps wont help so much with 3-7mm. When you clamp it around the edge of the board, being so thin it tends to bow up in the centre. If you have areas of the cut that will be cut right out and are not part of the final part then you can use wood screws and screw it down to the waste board. This way you have a few points around the thin sheet that will hold the middle down.Then just some screws around the edge. It’s far faster than bothering with clamps and also you don’t have to worry about programming the machine to avoid the clamps.
We will have some clamps again in around 2 weeks.