TND, I'm fairly new with tenkara myself but you need a couple of things to make things easier for your next outing. Trust me on this one, I've done the very same thing at some point comparing tenkara to my western fly rod and reel. I just wanted to make sure I got everything covered. I first thought of the limitation of not having the extra line to play a fish, thinking tenkara can't possibly compensate well enough for that. So, I brought both on some of my outings.
First thing you need is to get one of them quiver or you can make one yourself. Make sure its much longer than your tenkara stick, you'll see what I mean. Second, I agree that a hook keeper rig is a must to hold your line. As you get more usage with your tenkara, you'll notice that you don't necessarily need to have a kebari fly to catch fish tenkara style. Use your standard western fly patterns and you'll see that you'll catch more with it using your tenkara rod. Maybe someday I'll be using more of the kebari type flies but for now its hard to match the prowess of my western fly patterns, matching the hatch. For example, its hopper season right now in my neck of the woods, I'd be silly to choose a kebari fly over a simple hopper pattern if hoppers are what the fish are keying on. I just want to catch fish and tenkara makes it much easier to catch fish with flies, especially with dries. The long tenkara stick just makes it so much easier to get a drag free presentation and anybody who has fly fished long enough knows that is one of the most if not the most important key to catching more (trout)fish.
Here is my quiver. You can see that its too short for my western fly rod while my tenkara is being used
. You'll do this a couple of times and then you'll just be taking your tenkara stick. That's what I predict.