How to make multiple SAM web apps from the same code base?

I built a webapp for 1 client using AWS sam. The code lives on a single git account and whenever I have changes to push, I use the SAM CLI to deploy the new changes. The app has a website, with a backend db.

I recently deployed another version of the same app for a 2nd client. The 2nd app has a unique website, with a unique backend db. Now to deploy changes I have been using the SAM CLI to update each app, using a different config-env for each app.

How would I go about scaling this up to 10 clients with 10 websites and 10 backend dbs, whilst maintaining one single codebase, without the need to run the update script 10 times?