The phrase emphasizes that someone is considered completely unreliable, without a shred of trust. Ever since the scandal, the community doesn't trust him an inch.
Pro Tip 2/3
Negative feelings
Using the phrase can imply that you have negative emotions or a dislike towards someone. She lied so many times that I don't trust her an inch now.
Pro Tip 3/3
Informal expression
This phrase is suitable for casual conversations rather than formal writing or professional environments. In a meeting with friends, I might say I don't trust him an inch, but not at work.