You would need some kind of catch-all amendments stating that the enumeration of certain rights shall not be construed to deny others, and that the powers not delegated to the feds are reserved to the States or to the people. You could put them right at the end of the original amendments for emphasis as a closing statement of the Constitution.
But if we enacted those who would ever enforce them? The feds would probably treat them as if they didn't exist.
If you make them vague then it'll be easy to interpret them narrowly.
If you make them crystal clear, courts would presumably enforce them, like they have in the past.
The short response here is: How often do you see that happening in the US?
But in any case, note that I'm explaining what it was intended to do and what its meanings and implications are. Whether it is successful in achieving its goal is beside the point for this conversation.
Meaning: its point isn't to prevent access to real evidence. It's not an attempt to grant you privacy. It's an attempt to ensure justice is served correctly.
This is also why you lose that right when you're granted immunity. The state can force you to provide testimony in that case.
Corollary here is that it's actually quite surprising courts are willing to side with the accused here. It's probably only a matter of time before rulings come to the contrary. If you care about privacy as a human right, you really need another amendment to make it solid.