What are the implications of a huge amount of unused hash power? A not unthinkable scenario is that the Bitcoin price decreases to, say, 10-20% of today's value. This would make mining less profitable, and the difficulty should adapt accordingly.

This should lead to a large amount of unused hash power, that could be made available to some majority attacker. 

What are the implications of just knowing that someone (although unlikely) could be covertly using this hash power to mine a parallel and longer blockchain before difficulty adjusts, causing all sorts of trouble?

Can this scenario be remedied somehow by altering the protocol?
I grasp your question, mining becomes unprofitable so mining hardware is turned off and the difficulty reduces. Then, an attacker coerces miners to turn on their hardware in a bid to attack the network.

This would be difficult to achieve. First, you would have to identify where the mining hardware that is turned off is, identify who it is that operated it and, make contact with them. Then, you would need to convince them to go along with your attack.

As mining is largely anonymous, at best you could potentially source some of the IP's that miners are connecting through. Nothing says that the IP a miner is connecting through is their IP and even less identifies the mining operator working behind a particular IP, and still convince enough to go along with your attack to make up 51% of the hash power. This attack is infeasible.