I personally find that fishing jigs in the grass like you describe is less successful than fishing them around rocks, brush or more open banks. When trying to get in under the matt, I will usually go with a T-rigged bait (worm, beaver, speed craw) - but I find the jig gets hung up too easily in the grass.
If you do fish grass, be sure you don't work them too much - just shake 'em a little like you are hung up while letting it sit on the bottom. Works some times.