That's going to have to involve sending your coins to a completely different set of HDKeys because if the curve is also broken in addition to ECDSA, then just signing all local transactions with the new signature algorithm won't be enough.
I'm pretty sure that's what Satoshi was saying in that quote - the software would automatically send all your money to the new address type we end up with. As ranochigo points out above, I don't think it is possible to leave coins on current addresses but transition to some form of "hardened ECDSA".
Interesting to see what that would look like though. ECDSA in bitcoin hasn't been exploited yet so it's not possible to pinpoint exactly the weak spots, but I'd assume that we'd have to break ECDSA public key recovery in the process - as in, you could sign the transaction using XYZ quantum-safe signatures, and then sign that with the legacy ECDSA signatures and broadcast that. The transaction can still be verified as correct by everyone, but you can no longer get the public key back, like Schnorr signatures.