I know alot of choices have been mentioned already. But what kind of jig fishing are you doing? This makes a big difference, flipping, skipping, football jig, finesse? 90% of flipping bites you either pick up and they are there or you see you line twitch never actually feeling the bite even with braid.

I have NRX, GLX, Dobyns, castaway, falcon and duckett rods. Have been fishing for 40 years, the 2 things I have found is the shorter a rod is and the more limber it is the more sensitive it is. This helps by putting the balance closer to reel, the more balanced the more sensitive

Finess jig- 6'10 medium action
Skipping- 6'10 medium heavy fast tip
Football 7'6 medium heavy
Flipping - Heavy fast tip

Since the OP specified the most sensitive, G.Loomis NRX or NRX+ stand alone at the top of the heap.

I have some Falcon rods and they are excellent. I bought them around the time Falcon was starting up, they are jig and worm rods and the feel is ok on them, just looking for something better. The rods I have would probably be the equivelent of the Low Rider in quality, these were their premium rods in their day. Brannan's Bass Shop is a dealer for Falcon. I've already looked at them there. May have to go back.......

I have used the Steez Bottom Contact, Kistler Z-Bone, and NRX and in my hands, the NRX is the most sensitive. Since you want only 1 rod, if you consider the NRX, I would look at their lure ratings and make sure they are close to the weight range you plan to use it for (they tend to be narrower than other brands). Another nice thing about NRX rods is they keep their value pretty well and they are easy to sell if you don’t like them.

The Shimano Adrena is right along there with the NRX for sensitivity.

The quality of the graphite used in the rod makes the most difference in the sensitivity.

After looking at some of these rods that is what I picked up on, it's going to be much broader than one rod. I'll mainly be casting a jig or Texas rig with what I'll be looking for but I also finesse and flip and pitch jigs as well. Most everything I use is less than 7 foot, most are old school 6"6", which was considered long in their day. because most everything else being made at the time was south of 6 foot. On the feel I've actually thought about what you are saying as well. Most of the fish I caught on a jig last year the bite felt like I was going through a clump of grass. I set the hook and it was a fish. I had few that were a sure enough thump and one or two that litterally hit like they were hitting a run away crankbait on the fall.

St. Croix Legend Extreme- used for first time last Sunday. Relly impressed

I found out Brannan's has some of the Cara Falcon's in stock. I may run up there this weekend and then go down to Sulphur Springs and compare that to what Dobyn's has.

I have fished Falcon rods since they first came out. LowRiders then Caras and they are still my go to rods, but G Loomis is much lighter and much more sensitive I think. I have bought 3 Loomis (1 GLX and 2 GCX) for pitching/skipping and they are fabulous. Light and very sensitive. My opinion

Power tackle the one

As noted by Capt Bryan, there really never is "one" rod unfortunately. Type of bait, presentation, wieght, even line all have an effect.

Personally for me, I'm a Dobyns guy, but Falcon Cara line is very good as well and if I ever switched, that is where I would likely go. My current setup (Florida based, tend to go heavier)
DX795 - Flipping/Pitching up to 1-1/4 and some punching (but will be adding an 805 for a dedicated Punch rod to handle 1-2oz punching setups).
DX746 - 3/4-1oz Jig-n-Pig type presentations
DX745 - 1/2-5/8 creature bait/flipping tube/that sort of thing..

Not sure what this finesse jig deal that keeps getting mentioned though... .

Dobyns DX 784...or Megabass Braillist Orochi...both excellent options