How to add new column to existing table and set it value based on existing column using EF migrations

No really good way to do that. You need to edit the generated migration manually as follows:

1. Modify the AddColumn command to create the column initially with nullable set to true (i.e. optional).
2. Populate the column using the Sql method and raw SQL commands.
3. Add AlterColumn command to change the column nullable to the intended false (i.e. required).

Something like this:

protected override void Up(MigrationBuilder migrationBuilder)
{
    migrationBuilder.AddColumn<string>(
        name: "Manufacturer",
        table: "Cars",
        nullable: true);

    migrationBuilder.Sql(
@"UPDATE Cars SET Manufacturer = CASE CarModel
  WHEN 'Passat' THEN 'VW'
  WHEN 'Land Cruiser' THEN 'Toyota'
  ELSE 'Other' END");

    migrationBuilder.AlterColumn<string>(
        name: "Manufacturer",
        table: "Cars",
        nullable: false,
        oldClrType: typeof(string),
        oldNullable: true);
}