Car acts dead after it gets hot...

Del, as usual, has you covered.
The only thing I'd like to add is that the starter relay is particularly robust.
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I changed the location of mine to the passenger frame rail up close to the firewall, where my cable from the battery-in-the-trunk terminates. From there I cabled over to the starter. Then brought the signal wires over to the relay.
Now this location is only a few inches from the headers, and there is no way you can get booster cables down there, and not even your hands until the headers cool off. So hot-wiring the car or even installing a bump-starter tool is a real challenge.
The thing is, the relay I installed in year 1999 came out of my recovery box, and could have been new in 1968. IDK. And it is still down there, and still works. That is why I say, that IMO, it is particularly robust.
I wanna say that maybe they are not all as robust as mine, but I wouldn't say that mine is a one-of-a kind, altho, RRR might have something to say about that.
I have taken those apart, just as I take everything apart, just to see how it works. I been doing that since I was old enough to sit up by myself, much to the chagrin of my now poor dead mother. I can tell you that the guts look to me to be pretty tough.
As to why the thing works cold and not hot, I can only agree with Del, that you better check your battery connections and engine ground.

And in conclusion, I'll give you what I think, is a really great clue;
I once had a 92 Dakota that kept doing that, but quit after I removed the starter, and cleaned the starter to BH interface.
Same on my 1980 Volare. Now that I think of it, I'm starting to remember other cases...... Aluminum "rusts" differently than iron, and it seems that even a fine layer of oxidation can be troublesome.
Happy HotRodding

BTW, there is an easy diagnosis for this, but you need a helper. Have the helper turn the key to start and hold it there, while you are under the car. Take a hammer along and tap the starter up near the BH interface. If it springs to life, either the starter has a bad internal connection, or the ground is bad. With the starter system now known to work, from this point; clamp a jumper cable on the starter, and take it back to the battery; if it works now, then clean the interface.

If you want a short cut, just do the jumper cable thing first, and hit the key yourself. I just like smacking starters