I think an additional layer may be indeed a good option, but to solve the problem which is described in the OP, it should be a proof-of-work layer with tail emission.

We could have two different models which already in part exist today:

- the Namecoin/"Dogelitecoin" model: Blockchains which store transactions of a completely independent cryptocurrency, but can be merged mined with the main cryptocurrency ("Dogelitecoin" refers to Dogecoin being merge mined with Litecoin, which enhances both chain's security significantly, while Namecoin is still merge-mined with Bitcoin but the security improvement is quite small now due to NMC's low value/marketcap)
- the Drivechain / pegged sidechain model: a sidechain whose main purpose is to store and process Bitcoin transactions, or more precisely, transactions of a token pegged 1:1 to Bitcoin via two-way peg. The layer would have an auxiliary utility token merge-mined with bitcoin, with tail emission.

A proof-of-stake layer would not enhance Bitcoin's security at all, because miners and stakers are not the same group, and proof-of-stake validation of a sidechain doesn't protect at all against layer-1 51% attacks. In contrast, the merge-mining model does improve it, as it ensures additional income for miners providing funds to increase the hashrate.

If what you mean is a PoW/PoS hybrid Layer-1 blockchain the situation changes perhaps (there were several proposals, including Ethereum's Casper FFG and the even older "Slasher" protocol), but PoS depends heavily on the "weak subjectivity" paradigm, which means that new participants need a trusted source for their first connection to the network, weakening the decentralization of the whole system. PoW + PoS isn't a simple equation, as PoS provides another "type" of security than PoW.