My experience has been that burley needs to rest for a few weeks to months after coming out of the kiln. It's smoking qualities change significantly during that time.
Burley has a peculiar trait in its chemistry. All tobacco has two major tasks to complete during curing:
- breakdown of the chlorophyll while still alive (the color change)
- breakdown carbohydrates, proteins and other complex compounds that detract from its smoking qualities
For most tobacco varieties, these two processes occur simultaneously, so that once the leaf is brown, the second set of tasks is also mostly complete. This is not true of many burleys. Those that carry the genes for defective chlorophyll metabolism (white-stem varieties), brown quite rapidly, but the other biochemical processes are not synchronized, and require more time than the color change. So, when burley first becomes fully browned, it's not finished curing.
I would guess that burley is unsuited for sun-curing for just this reason. I usually kiln my color-cured burley starting around Thanksgiving (late November), to allow it time to finish its curing work. The best time to kiln burley is after the marked "raw" aroma softens to a mere "grassy" smell. Better yet, allow it to hang through the winter and spring, then kiln it.
I'm not aware of any traditional methods of handling burley in which it is sun-cured.
For your batch of sun-cured and kilned burley, I would suggest just putting it away in low case, to rest for a few months.
Bob