Starter Switch Fail (or something)

This is a Mercury Sable 2001, Vulcan engine 3.0L, Sedan body
The electric of the car functions normally except occasionally, the ignition switch doesn't work properly.
When the key is in the run position, all the accessories work normally.
When the key is in the start position, nothing happens.
I sometimes can turn the key back and forth, and then the switch works normally.
Once started, things are fine.

I replaced the neutral safety switch, and nothing changed. This was some months ago.
Later I changed the electrical part of the ignition switch, the part under the column on the left (driver's POV)

Still, the problem recured occasionally

Since the car usually starts with playing with the key, I just lived with the problem. Now, it seems to have gotten worse.
When the problem did occur once, I jumped the starter, and it turned over properly, but the engine didn't start.
To make the repair difficult, I can't cause the problem at will. Usually, everything will work fine.

So, what's my next troubleshooting step? Should I just change the ignition switch again? Ideas?

Sounds like a worn switch to me. We had similar issues on my cousin's Civic and replacing the ignition switch solved it.

Sounds like a worn switch to me. We had similar issues on my cousin's Civic and replacing the ignition switch solved it.
As I said, I already replaced the switch, the electrical part, that is. Today, I bought another one in the hope the previous one I bought was somehow defective. $21 at Autozone.
I really need a schematic for the car. There must be one on the web somewhere. Does anybody have a link?

You need to replace the whole switch. It sounds like the tumbler is just flat out worn out.

You need to replace the whole switch. It sounds like the tumbler is just flat out worn out.
What's the tumbler? My understanding is that there is a 'lock cylinder' and some kind of linkage going down to the switch, which is separate from the cylinder. Where is the tumbler part? Does that wear?
Perhaps I should mention that occasionally, after inserting the key and trying to turn, the cylinder is blocked from turning. But playing with the key a bit always overcomes the trouble.
I'm thinking of getting a new cylinder. There can't be too many things involved here.

The tumbler is part of the cylinder.

The tumbler is part of the cylinder.

https://lh3.googleusercontent.com/5LL_MIOO648ZzaE_GEg621TqqxvQ_uzYLf7xDpwHVV6wbJ--q7qFZ_Y7JQ8Wl62hm84xFF187wwhO8z22vyrVT_J3iVAXFwTgm 3-qXnZGOsO-1ERU0F06XNwdYqAP1RUTfNpk6LJP2f-FyH0psDwkPRznAhzzNhL1yx7eTv79GHGc4jwcR8-yYUFSq2OejsrY3IRq0XhZUqlKvjfVXUA8V8BOqU6kTFkbE1PiW 5vAxJd_JnJ-SYhEc-2arbCULEMqv7PVU9lm4QPym3S1choVajQmG99v1MUzO3D5Gs0Q UHpbMxHv4cmIgCJYMzvQNPD4jwaCArIfwWZbiWqW6VgXMUvDhI A_L_XmnxT4AAJ8rGsVMFgkOQqRzxyfTK-CDLu-gVaQJaNCYZBA-75tAQ1dGrhHZ-QIJsSONqJ-aGxmf6HBRO3kM8lZoViRZAf-TIkuS_XSI89poxVi0a0qonY8olTDdl8pntdY2OZp4rZNVhAbIB ngbwTZ3vKSltMgx8-YxibZaQPShpDGrTe5rHn2RAfWJhP_HirAAoUjgwnRlo=w1006-h511-no
I removed the old cylinder for inspection. I can't see anything wrong with it. The end paddle part seems perfect. It's about $30 USD to replace. I suppose that's not much even if it is the wrong part.

(I'm OP)
Although, parts stores sell both a transponder version of the lock cylinder and a non-transponder version (for this year and make), the counter guy I spoke with said that only the transponder version would work in a car that previously had one. Therefore, assuming the lock cylinder is the problem, I need to get the transponder version, which is about $80 USD. (http://shop.advanceautoparts.com/p/dorman-oe-solutions-ignition-switch-lock-cylinder-924-724/10609118-p?cm_mmc=PLA-_-Google-_-GPLA-_-10609118&ci_src=17588969&ci_sku=10609118&ci_gpa=pla&ci_kw=&iv_=__iv_p_1_a_214327102_g_12425515822_w_kwd-73682442382_h_1017527_ii__d_c_v__n_g_x_pla_y_62016 84_f_online_o_10609118_z_US_i_en_j_73682442382_s__ vi__&gclid=Cj0KEQiApruyBRCFqoDu1pbk9rkBEiQAF8EFdd4pbqVl Ur0s8xjeLiOQRgn3TA6RijhMU2PqaMMvi9IaAnc88P8HAQ#utm _source=pla&utm_medium=google&utm_campaign=gpla&utm_content=10609118)

http://shop.advanceautoparts.com/wcsstore/CVWEB/staticproductimage//521/large/10609118_dor_924724_pri_larg.jpg

I'll buy the part tomorrow; I've already ordered it from O'reilly's, but I'm still not sure it's the cause of the problem. So far, I've changed the neutral safety switch and the ignition switch (electrical part), so replacing the lock cylinder is a reasonable way to move forward. However, it's still possible the problem is caused by something else. I wonder about the linkage from the lock cylinder to the ignition switch. I'm not sure what's going on there.

Didn't work. Replacing the lock cylinder actually made the problem worse. Just before I did the replacement, I tried the ignition and it worked. (That's the problem; sometimes it works and sometimes not.) I replaced the lock cylinder and tried turning the new key. It failed. The engine did not start.
https://goo.gl/Q9lBCr
I returned the old lock cylinder to its place, and it worked.
I had some difficulty taking out the new cylinder, so I removed the plastic steering column bottom cover (I made a cut above where the key goes; see pic) to get a better look.
The lock cylinder is unfastened by depressing the pin in the bottom through a hole in the lock cylinder casing.
Anyway, although I'm so far not successful, I can now suspect the problem has something to to with the electrical connection to the lock cylinder.

I hope I can return the new lock cylinder. It's from the local O'Reilly and they may be fussy about accepting returns of electrical parts.

https://goo.gl/SsccGq
This is the steering column from below with the plastic bottom cover removed. In the center of the pic, between the wire and the shadow of the wire, you can see the hole that accesses the pin in the cylinder surround that holds the cylinder in place. This is the pin that must be pushed up to remove the lock cylinder.
In the previous post, you can see the plastic cover. I have a phillips screwdriver going through the hole in the plastic cover that reveals the hole in the cylinder surround.

(I'm OP)
If you haven't read the whole thread, I've concluded the ignition lock cylinder isn't faulty. The problem, which is occasionally not starting when the key is in the start position, apparently is something electrical. I theorize if I can get power to the engine with a jumper wire, that will work. Thus, I need to figure out which wire supplies the ignition electricity to the engine.

IOW, assuming the ignition lock cylinder is functioning properly, which wire is hot in the run position?

I really need a wiring diagram for this car. Anybody have one?

It appears the primary problem was the battery in the transponder-like key head. I replaced it, and the problem hasn't reoccurred.
I'm using an aftermarket key that replaces a transponder key. It has a replaceable battery unlike the Ford transponder key.