You can SSH to a host inside a private subnet directly by combining the IPs/hostnames with a plus: ssh 54.85.96.204.aws+10.0.1.96

The first IP is added a .aws because I want AWS IPs to trigger usage of an AWS specific config section in ~/.ssh/config that uses a proxy and doesn’t clog up my known_hosts file with SSH host keys of disposable machines.

These are the entries in my ~/.ssh/config that allow the above command to get you straight to your machine: