Vouches and references are helpful in gauging a candidate's integrity, but I want to learn of other approaches that might help filter out people who lack integrity.
The definition of integrity is subjective but that's probably beyond the scope of this post.
-- Warren Buffet
This is what your post is based on, right?
Some managers will be better judges of character than others but as soon as you formalize the process, you also very likely open it up to be exploited by people with low integrity and high intelligence.
- Discuss a time when your integrity was challenged.
-Tell me about a time when you experienced a loss for doing what is right.
- Tell me about a business situation when you felt honesty was inappropriate.
(If nothing else, ensuring that employees don't e.g. have massive debt makes it less likely that an otherwise-reliable person feels forced into doing something bad...)