password-less ssh into multiple machines from a single machine

I have been trying to get password-less ssh login into two or more machines (servers). The tutorials that are available online are great but they do not cover one corner case.

The default private and public key are named id_rsa, so every time you attempt a password-less login, ssh looks for id_rsa private file and matches it with the ~/.ssh/authorized_keys . However, what happens when you want to login to multiple servers

1) put the same public key id_rsa.pub in all servers where you want password-less login
2) create a separate private-public key pair and use ssh-agent to add the private keys
for each public-private key pair (new_rsa and  new_rsa.pub) do the following

localuser@localmahine$scp ~/.ssh/new_rsa.pub username@server:~/.ssh/new_rsa.pub # copy to .ssh folder
localuser@localmahine$ssh username@server # login to the server
username@server$cat ~/.ssh/new_rsa.pub >> ~/.ssh/authorized_keys  # append to existing aurhotized keys


localuser@localmahine$ssh-agent bash
localuser@localmahine$ssh-add ~/.ssh/new_rsa