Starting problems

[jimmyj]

New starter, new alternator, the battery is less than a year old and the distributor looks relatively new (but I'm not sure). Looks like the previous owner -- from whom I bought my awesome '72 Super Cheyenne C20 350 on Tuesday last week -- might have had the same problem I'm experiencing and started replacing these parts one-by-one in an effort to get to the bottom of things. Here's the deal:

First of all, it will start up strong in the morning, but despite pumping the gas it will inevitably die after 1-2 seconds. It will do this two, perhaps three, times before she lets me rev her up and get going; the colder the morning, the more times I have to turn the key and pump the gas before it finally wakes up (never more than four times, though). Then for the rest of the day, it's fine -- starts up strong, keeps on going, never stalls out or dies when I'm at stop lights. Purrs like a kitten. But if I drive for a while and then turn it off only to turn it on again within a few short minutes, the key won't even turn over. No click, nothing. If I wait about five minutes, it starts up strong again. This has happened approx. four times in the last five days.

Any ideas? Carburetor? Wiring? I'm sort of a novice, but I'm starting to get the hang of it. I'm in love with this truck & I need to get it into decent fighting shape so I can use it for the consruction company I just started. It's all I've got!