Cutting tiny jump rings can be a challenge. The action of sawing makes the coil move around quite uncomfortably in your hand as it acts as a clamp. I have personally tried several solutions, including holding the coil in special jump ring sawing pliers and using masking tape to prevent the coil from skewing and moving around. None were really efficient and I usually ended up with sore fingertips.
After coiling the wire onto a mandrel, use the same mandrel to hold the coil as you saw. You want the placement of the coil to be almost, but not quite right up to the end of the mandrel. This will keep everything as steady as possible as you saw. You’ll hold the mandrel/coil in your non-dominant hand.
If you don’t have one already, file or saw out a small groove into your bench pin (see the photo).
Rest the coil in the groove and begin sawing. You’ll slowly feed the coil up the mandrel as the jump rings are cut.
Pro tip: Put a tray of some kind under your bench pin to catch the rings as they fall off the coil.