My response is guess/speculation for this question but I have seen this in the past related to cold weather.
The starter gear is combined with a sprag clutch......a sprag is a one way gear.....or a one way clutch.
The purpose is so that when the engine starts, the still engaged starter pinion gear, disengages from the sprag, allowing the gear to spin faster than the starter motor shaft. This is to prevent overspeeding the armature of the motor which could cause the armature wire windings to be torn off due to centrifical (or centrifugal....memory blank) force.
The sprag is made up of several rollers that ride on the shaft and when the shaft turns, the rollers 'jam' the shaft to the starter pinion, causing it to turn. When the pinion begins to turn faster (the flywheel of the now started engine is turning it) than the shaft, the rollers release it from the pinion.....hence, a one way drive.
In the OP's case, there may be grease that has migrated into the sprag housing, holding the rollers in a 'released' position.
So, even though the solenoid has engaged the starter to the flywheel and motor power has been applied, the motor just spins without engaging the pinion gear.
Multiple attemps eventually warm the starter and the slightly greasy rollers free up and engage the pinion.....the problem majically disappears.
In my era, this type of description would likely having me replacing the starter drive if the starter was young enough with a fair chance of replacing the brushes in conjuntion.
Since starter repair parts are no longer marketed (except to formal rebuilders) this vehicle would get its starter replaced since we know there is no way to repair the unit, and the problem will likely recur.
Hows that for a long winded description of what might be the OP's concern? First thing on a Sunday morning no less.