I doubt the splitting is due to the heat. Unless it got wet after digging I think it's because you harvested a bit late. Sometimes the garlic won't show it at harvest, but will do it while curing.
It sounds like you're planting garlic in the same spot as last season. That's a recipe for disaster. You really need to find a spot to rotate it each year, not planting in the same area for at least 2 years, preferably 4. Garlic diseases creep up and can cause serious problems if you don't rotate properly.
As far as cover cropping, I think it's a great idea unless it'll interfere with the garlic.

-Mark

I agree with Madroneb.; Harvested too late.

You need to have some minimum of green leaves, to dry up after harvest to make some strong wraps.

Thanks for the information. We do practice crop rotation and each year so far we have planted the garlic in a new spot. Now we have a larger area that has better soil so we're going to plant there next year. Since we laid the garlic on the wagon with just a shade cloth over it, it probably did get wet. Next year we will put it in the shed. It is very windy and the sun is very strong here, the garlic leaves are usually dried up by mid June and we don't harvest until the end of July. Is that to late?

Yes, once the leaves are dried up it is too late for harvest. The recommendation (and what has worked best for me) is to harvest when 4 or so of the bottom leaves are dry, but the top 5 or so leaves are still green.

Then cure out of the sun (not just under a shade cloth) and out of the rain -- your shed sounds perfect, as long as the sunshine doesn't get in.

Store in a dry place. Come spring, make sure it's in a dark place so it doesn't get any ideas about sprouting.(I braid mine and hang in a closet)

I've harvested late before, and cured under a shade cloth in the sun, and ended up with garlic that was dried up before the next crop came in. Done as above, my garlic is still fresh when the next harvest comes in.