Zone 6, southern west-central KY here - about 70 mi NW of Nashville, TN. Illinois Everbearing produces for about 6 weeks, from early June to mid-July.
Not sure about Kokuso - just have a branch or two of it grafted into a fenceline M.alba, and it's only been fruiting for a year or two, so I don't have a good handle on it.

I think most of the variation in ripening times is along species lines, leaving most named cultivars grown in northern climates ripening around the same time, late June being the heart of mulberry season in every climate I've picked them (central valley of CA, desert SW, NE states, CO foothills). That said, Morus alba tends to ripen earliest (May in many places)and Morus nigra the latest(typically into August).

I've been wanting I.E. for a long time, but now that my Black Beauty M. nigras have come into bearing it's hard for me to get motivated to buy any other mulberries, since the latter is so much better than everything else I've tasted! It's also more cold hardy than most sources claim, but may need more sun and heat than Northern and Eastern locations have.