How can one send commands to the "inner" ssh session?

Add another tilde (ie, type ~~.). Each successive tilde is eaten by the outermost ssh session which hasn't yet eaten one, but if the next character is another tilde, it's passed along to the next session in.

If, from gromit1, you ssh'ed to a third host (let's call it wensleydale), then ~~~. would drop the session to wensleydale and return you to a prompt on gromit.

1 And what a great server that is; how often have I heard a developer remark "cracking host, gromit"?