1997 Ford Explorer v6 ohv weird 2nd start issue

So, I started my 1997 explorer up the other morning.  I stopped it after a
minute or so as I had to jump out at the end of my driveway to check
something.  I jumped back in.....no start.  Cranks like nobody’s business.  I
abandoned ship and took another vehicle.  A couple days later, dead cold
vehicle, it starts right up.  I threw a crank position sensor at it anyways,
because they are cheap, and I had no codes to go off of.  If I shut it off rather
quickly, the same thing happens.  Cranks great, but no start.  If I let it sit, and
get dead cold, it fires right up.  If I warm it up it starts up repeatedly without a
hitch.  It’s just that 2nd cold start that is not so good.  Upon further fiddling, if
I floor the accelerator during the 2nd cold start it will fire up.  I have good
spark at all times, and 30 psi of fuel pressure.  I think it is something weird
with a sensor that is messing with the injectors during the 2nd cold start.  Any
ideas?