Odd cold start problem with 1997 Hyundai Accent

On a coldish day after not having driven for a few days, turn the key and nothing happens. No crank, no fire. First impression is that I left the lights on the last time out… but then, when I let up on the key (bring it from ‘start’ to ‘run’), the fan and the radio come on, so it’s clear that the battery’s not completely dead. I try a few more times with the same result, start wondering if the starter’s gone, or the ignition switch.

Just for kicks, I put the car in neutral (it’s a 5 speed manual). This time, there’s a hesitation once the key is in the start position, but after that split second the engine cranks and fires, and sounds completely normal doing so. Drive three blocks to the gas station, during which it seems to run perfectly normally. Fill it up, and it starts back up with no problem (in gear this time, no hesitation). Drive 20 miles to the office and the car sits outside for several hours, then starts up like a charm. (weather in the mid-30s, this day). Next day, no trouble. A few days later, mid-20s and windy, the problem recurs, once again it’s as though the ignition switch isn’t being tripped until I put the car into neutral and even then it hesitated.

Any ideas? My going theory was that there was some problem with the safety switch that won’t let you start the car without stepping on the clutch… but I don’t know why, in that case, it would work in neutral and not in gear, and why it would resolve itself after such a short drive.

Incidentally, have never before had a cold start problem with this car, even in below-zero weather, it starts with a minimum of cranking.

Appreciate any suggestions!

Power (12 volts) comes from the fuse, goes through the ignition switch, through the clutch safety switch, through the starter relay, to the starter solenoid. The starter solenoid energizes and closes contacts which allow power (12 volts) to run the starter motor. The problem is one of these is sticking. Which one? I choose…the starter relay inside the passenger compartment. Your pick?
You’re supposed to step on the clutch when you start the engine, or when not moving with the transmission in gear. What would happen if you could start the car with the transmission in gear?