Adding front page splash/landing to existing site?

Hello – any idea how to add a popup or splash/landing page to an existing site so that if they click on a button like “I’m a Doctor” they go to the main page, but if they click on “I’m a Patient” they go to a totally different site/url?

I don’t want to change the main front page as this might negatively impact search engine indexing that already exists for this site.

Any ideas?