That will depend on the condition of your suspension. If you have any sag, then best stick with 37"
If your suspension is in great shape, then 39.5" is the max. Interco made several tires in that size specifically because that was the largest that would fit on an H1, which was the exact same suspension setup as the HMMWV (at least the early ones were, no idea about the Alpha and whatnot)
The tire chains for the HMMWV that the military uses add almost an inch in every direction, which adds up to 39" diameter.