Global filter – is there a better solution?

I'm currently working on the design of a new version of our current SaaS and I am having an issue with a functionality specific to the SuperAdmin.

Our SuperAdmin, when setting up the environment for a new client or when doing customer support, needs to often switch from the view of one client to another. In the previous version of the software, we had one specific superadmin account for each client, which made the switch longer and more annoying for the user (the SuperAdmin has everytime to log out and then log in back again).

After interviewing the SuperAdmin and brainstorming together with him, we came up with a global filter always present (for this user role only) in the top header (see pictures below):

The user searches for the client and selects the company from the list, getting a filtered view of the platform.

However, now that the feature was implemented in our first beta, the SuperAdmin complained that this solution is not user-friendly. Since it's still in development, the search has to be optimized so I know it can be improved (for example, if you don't type a letter nothing appears).

Yet, I can't help but wonder if we made the wrong design choice.. what do you think? Is there a better way to handle this pain point?