If you have a huge organization with multiple sites, consider assigning one of your EAP staff to that location for training, communication, and relationship-building.
Other work sites make it less appropriate to just show up and do a walk through. In such instances, you’ll need to consider a specific reason to show up. Perhaps your moving posters, dropping of cards someplace, or simply having a monthly meeting with the site manager on what needs might exist for the organization's work unit.
Why does this external rubbing of elbows work? Employees see you, consider their personal problem (remember 12-18% of the workforce has one right now) and think about calling. Then, some do. Visit again in three months and different employees will phone. The key low-dose frequency of contact with these potential clients.
Employees are also reminded about the company’s investment in them. This is appreciated and the reputation of the EAP as “being available” grows. Hint: 800#'s don't have legs.