My 2004 Saturn l300 will not start after engine is hot

lots of reasons. 2.2L or 3.0L V6? no engine told and is and engine question , no VIN and if told we tell you what's under that hood.
Driving to KEY West FL. huh, LOL !!!!!
not told is does it CRANK ok fast or starter is dead.?
or cranks fast but will not start,. hot. << guess yes.
so in 18 years no tuneups done??? spark, filters ,belts ,fluids?

all spark parts bad, they hate humidity when bad, for 100years.
2.2L has 4 cops spark coils in large single unit. it may be bad
or just those old old spark plugs under it. and 4 boots bad too.
V6. uses same funny unified coil pack, but 2x3 and same exact answer packs , boots and spark plugs bad.
that is about it, but no starts have 50 causes and lots of tests to find
but a tune up is always good to do , (ignored a long time)

I'm guessing you had to remove all the coil to put the new one in. When you reinstalled the coil packs, did you make sure your they are grounded?

Saturn L-Series - Automatic Transmission Fluid Change ...

? 19:48
https://www.youtube.com/watch?v=X6Ga3MESkqo
Sep 13, 2013 - Uploaded by DJDevon3
Procedure for changing the ATF in Saturn L-Series vehicles that ... I HAVE A 2004 L300 AND I WAS MAD THAT THEY DID NOT PUT A DIP ...

Saturn L-Series Questions - how do i check transmission ...

P0403 is EGR (exhaust gas recirculation) circuit malfunction
P0727 is Engine Speed Input Circuit No Signal
http://www.check-engine-light.com/trouble-codes

the engine valves must be the culprit here, the input valve must not be moving up and down or maybe leaking causing the improper supply of fuel to the engine resulting in start-up and shut off problems.

if your ground is bad or has a short the cable then acts a reverse deducter when car is off try changing the ground wire or yhe light switch

check for loose drive belt could be broke belt tensioner noise if you hear heavy engine knocking noise you have bad engine bearings.

The starter and or solenoid on your Saturn L300 would need to be replaced. I would first replace the solenoid. When your engine heats up there is too much resistance to allow it to engage. I have seen this problem on 5 other saturns and the solenoid is very inexpensive. Hope this helps!

Sounds like a crankshaft position senso that is failing when hot. 

Check connections at battery cables,should be tight & not be able to move cables.Check for corrosion at battery terminals.Also I have seen problems at the inside fuse block where a terminal burns causing similar concerns as yours.Hope this helps