I'll throw something out as it's sort of related.
When I was doing a lot of firewood this year, it required a bunch of trips up-and-down a steep hill
almost 1/2 mile long. Well, the dog would follow on foot. After a couple trips, she would get exhausted and would ride with me. No problem on the way up but on the way down, she would constantly slide off the seat so I'd have to hold or strap her into place. All passengers have to wear seatbelts coming down the hill or you'll slide off as well.
Then I had an idea. I had some non-slip toolbox liners laying around. I put them on the seats and tucked them under. Dang things worked perfect. The dog could stand, sit or lay down without sliding. My butt didn't slide either.
This is similar to what I'm using (although this is a 24' roll and my pieces were probably 2'x1').
http://www.amazon.com/Duck-281876-N...8830896&sr=8-9&keywords=tool+box+drawer+liner
I'm thinking about getting one of these larger pieces and covering the seat. Not only for grip but to protect against many of those things that could tear the OEM vinyl. I wouldn't adhere it much, maybe a few staples on the loose ends (e.g. corners) so it could be removed easily for cleaning if/as needed.