It would seem multiple piece rods will suit your need. If you're a bass angler, it's going to be tough because most true "bass" rods are one piece. Anyway, if they don't break down, you might have difficult navigating through something as simple shrubs and trees.
I did make something that might work although it was not the original intent when I made it. It's ability to tote a rod was a by chance.
I devised it to help "free" the hand holding the rod to help me have two hands when releasing a fish that can't be taken out of the water legally and give me a sporting chance to take a photo.
Just take these items:
And put them together like this:
My son was gracious enough to pose for this shot.
The idea was that when I got that big steelie close enough, I could slide the rod into the pvc tube so that the hand that normally holds the rod could be free to take a picture of the fish, release it, or whatever else. I fish solo 99.99% of the time so I don't have a buddy to assist with any of those tasks.
As I mentioned, I soon discovered that it could be used to tote the rod in front of or behind you. It will need a slight modification to accommodate a multi-piece rod. I also know of someone who fashioned a similar contraption for his backpack.
Now if you were on foot, you might consider building yourself what I have dubbed my rod "quiver". A quiver that hods rods instead of arrows, lol.
Anyway, it could be made to hold however many rods you want to carry but my own experience as a bank angler has revealed that even when I carried 5 rods through my fingers (what a pain in the phanny), only 3 were used more than the other two. The rod quiver enables me to carry 3 rods in one hand and that is a major plus.
The same guy that made the backpack contraption took my low tech quiver up another notch.
Here it is in action, propped up on a bench, off the ground.
If I were to make one for a backpack, I would definitely be inclined to go the route shown in Gringo Pescador's picture. Nice job, GP! You got that right about trees. I almost placed straps on my quiver to put it on my back but then I learned despite being mindful how easy it was for the rods above me to get tangled up or hit things.