The photo you posted appears to be a home made side mount type carrier. The correct location for the spare tire on the 40 1/2 ton pickup is under the bed. Also note, the carrier unit for the 37 (and I presume the 38 and 39) is identical to the 40.
To get on the same page regarding nomenclature, the part I refer to as the bracket is the track way that bolts to the frame crossmembers under the bed. The part I call the carrier is the long piece that holds the tire up against the frame crossmember and slides in and out on the bracket.
The carrier is often missing from trucks, as it was somewhat cumbersome to get the spare tire down and back up into position, so, many of them were discarded and the spare tire was carried in the bed. The brackets usually survived as they were bolted to the frame and were not in the way of anything.
Do a search in the chat site forums with the name "Mark Yeamans" as the display name, and when the results are brought up, scroll down till you see the date 4/09/17. This is post #387328. In that post, you'll see eight photographs that I took of the spare tire bracket and carrier assembly. The first photo shows the complete assembly in place. The second photo shows just the bracket in place. The 4th photo shows an extra bracket I had. The 7th photo shows the carrier by itself. I believe that is the part you are looking for.
As ChevGene pointed out earlier, the parts catalog calls out three different carriers for 37 models, depending on wheel and tire size. I believe the carriers probably differed only slightly in dimensions so that they all worked with the same bracket and T-bolt.
I have one from a 39 I might sell,got it for my 37 but installed a Astro van cable system instead. I have the carrier and the part that attaches to the frame.
