Mitsubishi Mirage usually starts and runs ok but will not start again after warming up

91 Mirage starts and runs ok but usually will not start again after warming up. What are the possible causes and how do I check to eliminate dead end cures.

Get a spark tester.
When it doesn’t start, connect the spark tester to the coil and check for a bright blue spark.

Connect a fuel pressure guage to the rail.
When it doesn’t start, watch the gauge.

Checking spark is the fastest, easiest and cheapest thing to do.
It’s also possibly the first thing you should do.

Carry a bottle of water in the vehicle. The next time the engine won’t start when hot, open the hood and pour the bottle of water over fuel rail then try starting the engine. If the engine starts the problem is vapor lock.

Vapor lock occurs when the fuel in the fuel rail begins to boil from the engine heat as the vehicle sits. On a fuel injected engine this shouldn’t normally happen. However, if the anti drain-back valve on the fuel pump assembly fails where the fuel system no longer holds residual fuel pressure, and the fuel pressure drops to zero as soon as the engine is turned off, the fuel can start to boil.

Or you can connect a fuel pressure gauge to the fuel system, turn the ignition to the run position to build fuel pressure, and then observe how quickly the residual fuel pressure bleeds off.

Tester

When you say that it will not restart when warm, do you mean that it cranks normally but will not fire, or do you mean that the engine will not turn over?

if the starter will not crank the engine when warm, then either the starter is failing (likely) or something in the engine is binding when hot (unlikely).

If the car cranks but does not start until after it cools, it may be many things, but my first suspect would be one of the temperature sensors. A temperature sensor that chronically reports a cold temperature can trigger a rich cold start sequence, which floods the warm engine.