The gear position sensor for the Vee is inside the engine. It's a simple variable resistor, and over time, heat and wear and maybe dirt or varnish change the resistance values from what the system "expects" to see. As you noted, the OD light comes on early, and you can sometimes feel a slight change in the way the engine runs, since 6th gear uses slightly different mapping. There's also a speed restriction in 5th and 6th gears.
The easy solution is to plug in an Ivan's TRE. It gets rid of that damn stupid bright green &^%$#@! useless OD light, and any time you're not in neutral, it tells the FI system you're in fourth gear. The engine runs a bit better, and the speed restriction is gone.
If you have the clutch out anyway for some other reason, you could also try replacing the sensor. I don't see the point, really -- much easier to install an Ivan's TRE and be done with it.
2002 Suzuki DL1000 V-Strom, Dark Metallic Space Blue
1983 Suzuki GS850G, Cosmic Blue
2005 KLR685, Aztec Red - Turd II.2, the ReReTurdening