Engine 99 V6 Intermittent Starting Problems

My daughter's 99 V6 began having some intermittent starting problems roughly 3 months ago. The problem usually occurs after driving for a short distance before turning the car off and then trying to start it again, even if the car has only been off for a few minutes.

When we try to start it in this condition it goes like this:

1) Headlights, interior lights, door chimes, etc. work normally (headlights bright) until the key is turned to the start position.
2) When the key is turned to start there's a click or two from the solenoid and then all power goes out. It may come back within a few seconds or remain out indefinitely.
3) If I remove the negative battery cable or starter relay the power comes back (after reconnect of course) and it's back to step 1.
4) Once we resort to jump starting it always starts back up.

Things I've tried:

1) Cleaned up/tightened the battery cables at the terminals.
2) Had the alternator and battery tested at OReilly's. Both tested OK.
3) Replaced the battery anyway as it was getting old.
4) Replaced the starter relay as it was cheap.

I'm confused as to where to go from here. I've been told it may be the starter/solenoid or a short somewhere but if that's the case, why does a jump start work every time? I'd rather not remove the starter if i don't have to as I hear it's a major pain to get to the upper bolt. I've also thought about a power drain somewhere but the problem has yet to occur from just letting the car sit for a long time so this seems unlikely to me.

Thanks in advance for any ideas you may have!

Thanks for the response, Joe. The problem at the time turned out to be the positive battery connector. When I tightened it up the first time I failed to notice that the connection between the cable and the connector itself was actually bad. A new connector solved the problem for a time.

Unfortunately a new variant of the problem returned a few months later and I still haven't been able to get rid of it. Same as before, if we drive the car for 10 miles or so and then return home the starter will never work right until the car cools down completely. But now, even after the car is completely cool (as in the next day) it still might not start easily. Once it does start it seems to start again just fine every time until we actually drive it somewhere.

When the problem occurs, the symptom varies now. Sometimes there is no sound at all, sometimes the solenoid clicks, and sometimes it sounds like the bendix gear is only partially engaged and the starter is grinding against the flywheel. At least the power to everything else seems normal throughout, so I guess there has been some progress.

Here's what's been done so far with little to no effect.

1. Tried starting in neutral & adjusting the tilt of the wheel.
   
2. Tested all fuses.
3. Tested the battery for voltage & load.
4. Cleaned up all of chassis ground cables that I could find.
   
5. Replaced the Starter/solenoid (twice, once at a shop that assured me that the one I put in was no good).
6. Replaced the entire battery cable assembly with a new one from Ford.

I guess I need to test the S signal at the starter to see if either the transmission range sensor or the ignition switch is to blame but I'm not sure how I can do that since the solenoid connections are impossible to see once the starter is installed. Does anyone know of a good way to do this?

I tested the ignition switch and there was no continuity between P1 and P2 in Off/Lock, but otherwise it checked out ok. I can't find any info on exactly what this circuit controls, so I'm just replacing the switch. While I wait for it to arrive, does anyone know if a dying PCM can cause cranking problems? My code reader is having problems connecting when it used to connect and read codes just fine. It connects to my other vehicles just fine.