Jeff,
The 38 and up kick starters use a Sprag (one way) clutch in the starter drive device. I'm guessing the lubricant in the drive mechanism is stiffening in the cold weather, causing the Sprags to stick and not engage. If you take the starter motor off, you might be able to get some WD40 into the drive mechanism by spraying it into the narrow annular between the hub and the shield. If it frees up the Sprags, you should be able to hold the starter motor armature stationary and rotate the starter pinion gear freely in one direction, but not the other.
My parts manual only goes to 1952, so I can't say what the interchangeability is between the later model cars and trucks. Sometime around 1949, the cars went to a solenoid-operated starting mechanism, while the trucks retained the kick starter for a few more years. However, I think the clutch/drive mechanism is the same in both types.
I'd be interested in knowing what rear end you have installed in your 37 pickup, and how close the track width is to the stock 37 rear end, and what size wheels and tires you're using. You might want to PM me on that, as some people get a little edgy when the subject of modification from stock comes up.
Mark