Rubber ferrule for a tapered stick

My cane is tapered with a small rubber tip. It does not provide good traction on some surfaces. I would like to put a full rubber ferrule, but am concerned about the taper. I think that the difference in the diameter between the top of the ferrule and the tip would cause the cane to "wiggle" and possibly affect balance. Is this the case or would it be snug enough?. If not, I'd appreciate some tips on how I can fix this. The length of the cane now is perfect for me.