What should you do with your social media and SEO strategy when you have to rebrand your product? How about the website?

Some details…

We are launching a supplement and we manufactured the first batch of a couple of thousand units that are ready to go.

Based on some feedback and legal considerations we are opting for a different name and in the next batch, we are going to manufacture the product with the new name/label.

What should be the right strategy for social media, SEO strategy, short term, and long term, and also should we just keep the "old" website up with the old name and redirect it to the new website with the rebranded name? How about optimizing the content on both?