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How to add a second site to an existing jira account to create separate projects in the new site?
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You can have more than just one site under one organization (I think this is what you meant with "jira account") but the new site need to be transferred from one organization to your existing.
This is because a new organization in created when a cloud site is added.
Transfer products to another organization
As I understood you in a way you want to keep things absolutely separated this step is possibly not even necessary in your particular requirement.