The practice of Management By Walking Around is a great technique for executive directors and chief operating officers to develop better relationships with staff, understand their challenges, and learn of potential problems before they get too big.
By its name, it really is that simple - walk around your shelter and talk with staff members. Do this regularly but not on a schedule so that staff is not prepared for you. Often in the context of them doing their jobs, they will be primed to have more specific comments and ideas. Keep the conversation informal, don't be critical (that can come later if needed), and listen more than you talk. Also, acknowledge good work when you see it and follow up when people have questions or concerns. Following up builds trust.
For more details on why and how to practice Management By Walking Around, check out the PDF named "The Power of Taking a Walk" in the resources at right or at this link. It's an article Bonney Brown wrote for Animal Sheltering magazine.