I recently went to a show that was played by a Heart by Heart. It's a cover band that focus on playing Heart songs, but with the added appeal of having two of the original band's members: Steve Fossen and Michael Derosier.
A band isn't solely identified by the lead singer, so Heart by Heart having a different vocalist than the original Heart makes me feel like Cover Band is the wrong terminology. After all, two of the band's members are playing their own songs!
There is a list I found of famous spin-off musical groups or bands (not necessarily cover bands), where some members of a disbanded musical group later form a new one including some of their former band members.
Spin-off, sense 2 or 3 from Merriam Webster Dictionary:
2: a collateral or derived product or effect: BY-PRODUCT
3: something that is imitative or derivative of an earlier work, product, or establishment
This could be called a successor, that is, something that succeeds the original in the sense of (from Merriam-Webster):
1a : to come next after another in office or position or in possession of an estate
especially : to inherit sovereignty, rank, or title
b : to follow after another in order
I agree, that would not be a cover band. You could just say it's the new line-up of Heart by Heart.
