This is a great suggestion! A while ago Chaum, myself, and Mario Larangeira created the notion of quantum-secure fallbacks for wallets. First we started with actually hiding a post-quantum key behind the ecdsa key (easier to rollover, but not completely compatible with traditional wallets).
We now have a paper under review where we propose pretty much what you suggest here (which has already been discussed in a different thread) as well as the integration of that preimage into the ecdsa signature nonce to create a fail-stop signature scheme