One of the best tips I ever got for bicycle maintenance was one I read in these forums:

Ever notice how the first bead goes on easily? That's because you put the first section of the tire on the rim, and it doesn't seat... it falls into the bottom of the trench in the center of the rim. That gives you all of that slack to work with when you work your way around to the other side of the rim. Tire's first bead pops on easily.

Then you stuff the tube in and start popping the second bead on. Your tire will usually start to seat properly along the rim, and all of that slack that made the first bead go on easily disappears. The second bead is MUCH harder to force on.

Try this:

After you get the second bead on as far as you can... grab the tire/tube at the opposite side as that last difficult part. Squeeze the beads together and pull them down into that trench in the center of the rim as far as you can. Continue to squeeze and pull the beads as you work your way around the rim to that last tricky part of the tire. If you work your hands around and keep that pulling tension on the tire... it should give you some of that slack that you used to get that first bead into the rim. With this technique, I've always been able to pop the tire on without resorting to a tire tool or dish soap.