The new Illini canes with flowers forming sounds like what is called by
Harvey K. Hall a "Crown Lateral". It's a fruiting lateral originating near
the crown. It looks rather like a primocane coming up for the year, but it
flowers and overwinters as if it were a fruiting lateral. It happens to my
black raspberries sometimes if most of the floricane is damaged or cut before
growth starts in the spring. The only real test I know of to differentiate
crown lateral vs. fruiting primocane is to wait for next winter. If the cane
dies over the winter, it's a crown lateral. If the cane survives the winter,
it came up as a primocane. In the case of flowers are forming on a
non-primocane-fruiting variety like Illini, then it's likely to be a crown
lateral. Ripening season for crown lateral berries falls a few weeks later
than the end of the floricane berry ripening season. I've heard of a few
growers using the effect intentionally by mowing off the canes in the spring
to get off-season black raspberries, but I have not heard of it being
exploited for blackberries.