Well, we know that the societies of Slaver’s Bay were reconstructed after the Doom:
“What now remains of the once-proud empire of Old Ghis is a paltry thing—a few cities clinging like sores to Slaver’s Bay and another that pretends to be Old Ghis come again. For after the Doom came to Valyria, the cities of Slaver’s Bay were able to throw off the last of the Valyrian shackles, ruling themselves in truth rather than playing at it. And what remained of the Ghiscari swiftly reestablished their trade in slaves—though where once they won them by conquest, now they purchased and bred them.”
And certainly the rise of the Dothraki, and the way their khalasars’ raiding and migratory patterns “industrialized” the acquisition of captives which the cities of Slaver’s Bay would then “process” into skilled slaves for the Free Cities’ markets, would have probably contributed to an increase in supply of slaves.
However, we also know that the Valyrian Empire was a slave society stretching back thousands of years, so it’s not like Volantis et al. just started buying slaves for the first time. So the political-economy mystery here is what would have caused such a sharp increase in demand to meet the increase in supply.