Frustrating start problem

[rivcokid]

My 31 roadster is becoming more and more unreliable - so much so that I'm hesitant to take it out. Here's the issue: Sometimes it simply won't start on its own - by that I mean when I touch the starter rod, it sounds like the battery has been removed. Absolutely nothing happens. I can touch the rod 100 times and get the same result, but 2 days later, I can come back and it will start as usual. There seems to be no rhyme or reason to when it does this, except that I noticed it being much easier to start this morning with it being 95 degrees out. Anecdotally, it seems like the colder it is, the worse the problem.

When the car is in one of its "don't start" moods, I can start it with my jump pack with no problem - it starts right up with the 12-volt jump pack. I don't have to do anything to the car other than hook up the jump pack.

I've checked the battery both when the car will and will not start, and it is always good - 6.1 to 6.3 volts, and it will and won't start no matter where that number lies.
I've checked and rechecked the cables and connections, and there are no issues there. I have a secondary ground.
The starter spins very strongly - it was just rebuilt several months ago.
The generator indicates 7.2 volts.
I've removed and checked the starter switch and it looks fine.
There is a fuse on the starter and it is fine too.

In short, when the car works, it runs beautifully with no problems. When it won't start, it won't do anything until I hook up the jump pack - hence my hesitation to take it out now. There is no indication when it will change from wanting to start to not wanting to start.

Anyone have any other ideas? Thanks (again) in advance!