Experiment with different round-ended needles. Pins are sharp so tear at the putty rather than stretching it nicely. work in rows top to bottom or left to right pulling the putty away from the direction of travel. On the next line work bottom to top or right to left pulling in the opposite direction. Repeat across the whole surface.
Some tips-
The thickness of the putty layer is critical. Too thin a layer and it will tear, too thick and it's hard to get the loops nice and regular.
As well as pulling across, pull each row slightly over the one before so they overlap a little.
You'll find it more comfortable working in one direction than the other, so turn the model around instead of trying to work in the uncomfortable direction.
I find working top to bottom yields better results. That's not how chainmail is usually made though, so it's a toss up between effectiveness and realism.
Go around the hem afterwards opening up the bottom links.