My current SaaS runs on Heroku and it has custom subdomains for individual clients. I'm almost at a point of rebranding the effort and almost finalizing the new domain name. Wanted to ask the experts here if anyone knows of what it takes to rename an existing product that's hosted on Heroku which has custom subdomains?

Apart from updating the literal name everywhere within the product, I'm more concerned with what it takes to make sure the hosted code resolves to the new domain, and the transition involved in terms of code etc. I'm okay to refer the product within the code using the old product name,

Can someone please give me some tips on what to watch out for and if you know of, how to execute this rebranding effort? Your help on this is much appreciated.