No, a new publishing profile is not needed in order to change the publishing destination. In Visual Studio when you select the Publish option -> Specific Target -> Azure App Service, then a dropdown shows of resource groups in your subscription. Within the resource group for your app there should be a prod app service, this can be expanded to reveal a "Deployment Slots" folder.
If a publishing profile already exists, then when clicking the publish button the solution will directly start to build and upload, without an option to select the target. In this case, within the Solution Explorer on Visual Studio, there is a "Connected Services" section. When clicking on this, it takes you to a view with three options on the left: Overview, Connected Services, and Publish. Clicking into Publish, you can then select "Edit", and here is where the destination URL can be changed to that of the staging slot. For a target change within the same web app (just to a different slot), all fields can be left the same with the exception of the destination URL field. This field will be changed to appname-slotname.azurewebsites.net.