Increasing the size of tyres on a hybrid
For a 700c tyre you would be replacing it with a 29" MTB tyre.
The best way for you to determine what size tyre you can get to fit is to simply measure the gap between chain and seat stays as these are the most likely areas to cause problems with clearance.

Providing you don't want to ride anything too gnarly (i should hope not on a hybrid), a 700x35C cyclocross tyre could give you a much more grippy/knobby option without needing more clearance.  Something like this maybe?