When an administrator navigates to Experience Workspaces from Setup, they may be redirected to the Experience Site's login page instead of being taken directly to the Workspaces dashboard. This behavior also affects accessing the published site when navigating to the site URL via the link provided in Setup | Digital Experiences | All Sites, as well as from the App Launcher in Lightning Experience. In Salesforce Classic, the Global Header may not allow selection of a site to log into.
This occurs due to one or more of the following session and authentication configurations:
Each of these conditions prevents automatic pass-through authentication from Salesforce to the Experience Site, requiring a separate login.
Try the following options in order and verify whether the issue is resolved after each step.
Note: These steps apply only if you do not need separate (internal and external) sessions for your profile. If you do need separate sessions, proceed to Option 2.
Note: These settings are only available in Enterprise, Performance, Unlimited, and Developer Edition organizations. If your organization is not among these editions, you will not have access to this setting.
Salesforce Help - Modify Session Security Settings
000383785

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