That's just an easy way to create new hash functions out of an existing one. Nobody sensible would insist that your hashes must have different underlying algorithms, just that they produce uncorrelated outputs.
Haven't been reading is definitely true. Not sensible sounds like the case.
If you're using a cryptographic hash, your method clearly works. If you're using a fast and crappy hash function, you might need more care. But I suspect not.
As always, test with your real-world data. If it does what you need it to, theoretical qualms are likely irrelevant.