The cone is to stop foreign objects (including fingers) from falling directly into the grinder throat.
What you could do is, rather than dumping all your beans into the hopper at once, only add enough for your current shot(s). Then the beans arent in contact with the hopper as much. Also, the beans will stay fresher as well when not exposed to air and light as much.
If you havent already, get yourself a grinder brush. Dont use a brush with nylon or other synthetic hairs or you will get a static electricity problem with clumping
I suspect with your new grinder and machine, you may want to vary your bean choices. In fact, I highly recommend it.