This all makes sense. That's why Kaspersky offers free upgrade to a new version during a purchased license period. If the license had expired on the old version, you would certainly be "SOL." If not, you should be able to download the latest KIS version, apply that license code to it, and be happy for a year. I cannot confirm this latter observation, except to say that I'd always been able to upgrade to a newer version for free to run out the remainder of an active license period on an older version.
I must've said it elsewhere, but KIS costs a bit more than some other AV/IS programs. So we constantly keep abreast of new OEM "envelope" licensed versions and KIS download offers from reputable resellers, rather than pay the full price through Kaspersky-USA for license renewal. They apparently know and accept such practices, because you can install a new license code on a soon-expiring license, and the new license period of the new license code picks up only after the old one has run out.
I don't want to be an uncritical proponent of Kaspersky, but it has served us well for the good part of a decade. I started another thread on peculiar behaviors noted mid-license-period for a software upgrade released a month ago. You might want to read my discoveries about that, and the post I'm going to make on that thread in a minute.