Below: A map of Cursory LMLK Impressions (the earliest variety). Blue is Hebron, Orange is Ziph, Green is MMST, and purple are those with eroded inscriptions. As anyone can see, most are concentrated at Lachish (50), and there are some at Jerusalem/Ramat Rahel and Mizpah. Only four have been found at Gibeon. Curiously, the only lmlk handle excavated at Tel Beersheba is a Cursory Ziph handle on a large pithos. Beth-Shemesh, Timnah, and Gezer also contain Cursory lmlk impressions. Note the large amount of Ziphs, utter lack of Socohs, and paucity of MMSTs.
This strongly suggests that even at this early date, Lachish was effectively made capital of Judah by the fact of its strategic location, it being far closer to the the districts mentioned on the handles than was Jerusalem. I still cannot find a good explanation for the appearance of Beth-Shemesh, an unfortified oil-town, on this map. Perhaps it was a local administrative center.