In my opinion, the 'three section staff' is a variant staff. There really is very little you can do with one weapon that you can't do with the other. The three section staff potentially develops a more powerful strike at the expense of being much more difficult to control at all times and slower to recover from a far strike. I consider that mostly a wash except for the time required to learn the weapon. But basically, its just a variant staff.
That said, the staff itself is a very underused weapon which probably deserves its own feat tree (and I've seen at least a few attempts at this).
At most, you might develop a specific tactical feat for wielders of the 'three section staff'. If you look at the 10 primary techniques listed in the second link, you'll see that they are mostly basicly staff techniques or equivalents. Only two of them (the 'elbow strike' and the 'coiling dragon') have no real comparison with traditional staff technique and use the sectioned staffs flexibility, and those are the showiest most stage-like techniques. The best I can think of is that technique #1 'Iron Elbow' would a) grant a bonus on your Bluff check (say +4) and a small bonus to damage (say +2) when feinting with the weapon and b) Allow you to attack two separate targets (at -2 to hit) with a single standard action after a successful feint, and technique #2 'Coiled Dragon' would allow you to as a full round action make a single attack against a target 5-10' away in which your strength bonus was doubled.
So, putting that together.
Three Section Staff Technique [General, Fighter, Tactical]
You've learned the secret techniques of using the Sam-Jit-Gwun
Prerequisites: Proficiency with the Staff, Weapon Focus (Staff), 3 ranks Jump, 3 ranks Tumble
Benefit: If the staff you are wielding is of the three section variaty, you may use the following manuevers.
- 'Iron Elbow': When ever you feint, you gain a +4 bonus on your Bluff check. If you catch you opponent flat footed, you gain a +2 bonus to damage.
- 'Double Elbow': Whenever you have feinted, when you do your follow up attack as a standard action you may choose to do two strikes instead of one. Each strike is at your full BAB but at a -2 penalty to hit.
- 'Coiled Dragon': A a full round action, you may make a single attack at an opponent using the staff as either a normal or reach weapon. If the attack is successful, you may double your strength bonus for the purpose of determining damage.
Normal: A three section staff is considered to be a variant staff and gains no special benefit or penalty, provided that the weapon is on the users cultural weapon list.
As an aside, whenever you see that a weapon is 'rarely taught' you should understand that to mean that the weapon is difficult to learn compared to its effectiveness. If the weapon was truly effective, more people would have bothered to learn how to use it.