Or take a peek at the actual ignition switch, not the key cylinder.
Should be down low on the steering column. The key part drives a rod that engages the switch. It could be that with wear and tear, the switch is sticking and not fully releasing from "start" to "run".
Simple spring loaded plunger switch, the rod pushes the switch plunger to engage the start function, when you let off the key the switch plunger springs back to the run position.
And on that note, make sure the key is actually returning to the run position, when you let go of it.
If you want, shoot some PB Blaster/WD40 into the lock cylinder, where the key goes, and try working the cylinder back and forth a few times to see it it helps.
97 Sahara, 74 Renegade, 72 CJ5, 55 Willy's (the Beast) Boss 302, 6:34 gearing, Bose 901 speakers.
God I miss that Jeep. Forgot the kinda Jeep, 80 SX4.