I am very familiar with Entity Framework using Database First approach and how to deploy the solution with it.
Basically, in my solution with database first approach, I have a web client project that consumes data access library project that is coded with database first approach.
So, first, I write some SQL Server scripts to add new tables (or make schema changes).
Next, go to the data access library project, using EF edmx designer to update .net from existing database, compile this data access layer, and the DDL reference is automatically updated in the client web project.
When I deploy the solution to the production server:
If you're using Azure then you can configure it has automatic process as shown below.
Else you have to do it manually like this :
You have to create a db script and after that you can run it on your production db.
PM> Update-Database -Script
You can refer this doc : Getting a SQL Script