I'm trying to add multiple logins for my Stack Overflow account. In "My logins", I clicked "add more" and then when I click the "Stack Exchange" icon it's logging me into the existing account.
I'm not able to add another Stack Exchange login.
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You cannot be logged into two Stack Exchange OpenIDs at the same time. So, to make this work, you're going to need to add a third-party login first. Looking at the email address you have used to create your existing SE OpenID, I suggest adding a Google credential. :)
So, follow these steps: