What's the cranking speed like compared to a cold start? Slow cranking reduces the starter speed, which causes it to draw enough current to starve the ignition coil. The slower a starter armature turns, the more current it draws due to lack of something called "Counter EMF" which is generated in its windings when it spins in the magnetism developed by the field coils. If the starter can't spin fast enough for some reason, it draws so much current there's none left to power the ignition system. One dead giveaway that starter speed is the problem is the tendency of an engine to start, or at least fire a couple of times just as the starter pedal is released.
Jerry
