I work on a multi-tenant service that allows users to be members of multiple organizations. External users are typically only members of a small number of organizations, so the current Organization Picker works great for them – either they bypass the picker to log in to their single organization, or they select from a short list.
Internal users, on the other hand, often have access to well over 100 organizations. For example, our support staff needs to log in to organizations to understand problems reported by the external users. For them the current Organization Picker is inadequate, since it is only capable of displaying 20 organizations.
My suggestion is that the Organization Picker become a searchable drop-down list once the number of organizations exceeds 20, and include all of a user’s organizations. Typing a small part of the organization name to filter the list would allow them to easily select the organization they want and the drop-down control would occupy far less real estate on the form than 100+ individual buttons.