You'll need two DbContexts.

public class BloggingContext : DbContext
{
    public DbSet<Blog> Blogs { get; set; }
    public DbSet<Post> Posts { get; set; }
}

public class MyBloggingContext : BloggingContext
{

}

public class MyBackupBloggingContext : BloggingContext
{

}

And you can register them like this:

public void ConfigureServices(IServiceCollection services)
{
    services.AddDbContext<MyBloggingContext>(options =>
        options.UseSqlServer(Configuration.GetConnectionString("DefaultConnection")));

    services.AddDbContext<MyBackupBloggingContext>(options =>
        options.UseSqlServer(Configuration.GetConnectionString("BackupConnection")));

}