So, having detailed tests of internal functions will generally improve the chances that your test will catch a regression.
That's the general theory, and why people recommend unit tests instead of only the broader possible integration tests. But things are not that simple.
Interfaces do not only add data, they add constraints too. And constraints reduce your state space. You will want to cut your software over the smallest possible interface complexity you can find and test those, those pieces are what people originally called "unities". You don't want to test any high-complexity interface, those tests will harm development and almost never give you any useful information.
It's not even rare that your unities are composed of vertical cuts through your software, so you'll end up with only integration tests.
The good news is that this kind of partition is also optimal for understanding and writing code, so people have been practicing it for ages.