My daughter’s ’90 LE Auto 155,k mi had cold starter not work problem. After warm was always fine. We had replaced starter and battery a year earlier and so thought starter was still OK. But one year and several winter months of turning key w/o cranking at all, she took to private garage and they replaced with Remanf Bosch starter. Everything works fine since. Plus the Garage and Bosch guaranteed one year parts and Labor(?). Not bad price either; Bosch seems only remanufacture of these Toyota starters; Toyota will only sell new.
If warm starter does no crank/turn at all, may be internal wire winding shorting out after heat expands parts (but this usually happens due to cold condition). If cranks and does not fire, can be ignition coil defective, or gasoline flooding. Remember if does not fire after several (few) tries, way to proceed is keep gas pedal fully depressed while cranking. Of course, never depress or pump gas pedal when staring any Fuel Inject engine. Should be no need whatsoever to touch the gas pedal until engine fires up a little.
One last thing on the 4FE engine, there is a idle air-bypass screw/plug in the intake manifold right at the throttle plate housing (easy to see the screw’s slotted head inset in throttle body). I would check this for dirtiness. If you want the step by steps to do this, please reply and ask. I don’t what to complete with Haynes book.