Find the websites distributing it, infiltrate them, generally do the legwork to find what's going on - which is exactly what they've been doing.
"But the children!" is not a skeleton key for privacy, as far as I'm concerned.
I reject on-device scanning for anything in terms of personal content as a thing that should be done, so, no, I don't have a suggested way to securely accomplish privacy invasions of this nature.
I'm aware that they claim it will only be applied to iCloud based uploads, but I'm also aware that those limits rarely stand the test of governments with gag orders behind them, so if Apple is willing to deploy this functionality, I have to assume that, at some point, it will be used to scan all images on a device, against an ever growing database of "known badness" that cannot be evaluated to find out what's actually in it.
If there existed some way to independently have the database of hashes audited for what was in it, which is a nasty set of problems for images that are illegal to store, and to verify that the database on device only contained things in the canonical store, I might object slightly less, but... even then, the concept of scanning things on my private, encrypted device to identify badness is still incredibly objectionable.
In the battle between privacy and "We can catch all the criminals if we just know more," the government has been collecting huge amounts of data endlessly (see Snowden leaks for details), and yet hasn't proved that this is useful to prevent crimes. Given that, I am absolutely opposed to giving them more data to work with.
I would rather have 10 criminals go free than one innocent person go to prison, and I trust black box algorithms with that as far as I can throw the building they were written in.