How to rebuild brand trust after a data incident and social media misperceptions?

[MiaJ]

I'm part of a small PR team at a tech startup that recently faced a minor but public data incident, and while we resolved it technically, the online narrative has spiraled slightly beyond the facts, impacting our brand perception on social media. Our leadership wants to rebuild trust proactively. For communications professionals, what are the most effective steps beyond the standard apology statement for genuinely rehabilitating a corporate reputation after a misstep, and how do you measure the success of these efforts in terms of shifting sentiment and restoring stakeholder confidence over the medium term?