Most likely a bad connection
or
sometimes new starters are bad too, they're made by humans in some cases

good luck
kickstart

If it doesn't start after sitting in the garage overnight, then it's
not a heat-soak issue with the headers.

Did you also replace the starter solenoid? If you did....

Sounds like an electrical problem... should be easy enough to put a
voltmeter
on the solenoid to determine whether turning the ignition key to "start"
feeds any
current to the solenoid; sounds like it sometimes doesn't. Then it's a
matter
of tracing the wires to find the bad connection, or try bypassing the
starter lockout switch (probably on the auto tranny linkage, or behind
the clutch
if you have a manual transmission).

One thing I have found is that the bulkhead connector on the firewall can
give a problem like this, especially if the engine has been pulled where the
wiring harness has been moved around.

The pins in the connector will sometimes separate slightly when hot, giving
the hot soak symptoms of not starting hot.  The check is when it isn't
starting, put a test light on the solenoid wire (inner small terminal on the
solenoid) and have someone click the key.  If the light comes on, you have a
solenoid problem.  If the light doesn't come on, you have a wiring/bulkhead
connector problem.

Another similar issue is ground.  Especially after an engine change, engine
ground wires often get ignored and the engine will run fine most of the
time, but have starting issues occasionally, usually hot.  I don't know why,
but the quick way to check and fix, or eliminate, is take a jumper cable
from the block, intake, alternator to the frame and/or battery negative
post.

There is nothing more frustrating than starting problems.

If these aren't it, then there is the solenoid spring issue.

Chevy has a lightweight solenoid spring and Bob G has the part number. It is
like $5 from Chevy.

corroded battery terminals or where the ground attaches to the frame; maybe
the positive line to the starter dirty connection or the solenoid.  Less
likely but possible internal damage due to corrosion of the main battery
cables (especially POS at the battery).  Turn the headlights on if they are
bright and then dim to almost nothing as the ignition switch is moved to
START it's probably a dirty/corroded connection somewhere.  If the lights
stay bright it's the solenoid.  (some cars the headlights will cut off
during starting but I don't think that '78 does that maybe someone else that
knows that car will confirm)

Butcher
'96 LT-4 CE