I would bet your kingsnake loves to try to find a way out of the cage, and that is why he/she is climbing. Or, when hungry, the snake is cruising for food.
Kings are ground-dwelling creatures and in the wild spend most of their time hidden from view, below the surface.
In my mind, a better cage setup would be to have a nice temperature gradient (and that includes a cool area), alot of burrowable substrate, and a moisture source. I bet you will see less climbing attempts and more time in and under the surface.
It isn't that adding branches is bad, it may just be giving the snake more to climb on when he/she is looking to get out of the cage.
Make sure the snake is not too hot and trying to escape. Your cage right now should have a cool area for the snake to retreat too.
Good luck. And I have no idea about grape vine from your yard. The dried grape wood sold at stores and online is durable for terrariums, and is completely oven-dried before sold.