As pointed out in this Ask Ubuntu answer, you may use
zssh instead of
ssh. When using
zssh, you'll be able to switch to file transfer mode using Ctrl + @. This should allow you to transfer files back and forth between the local and remote ends of your active connection.
Another trick pointed out in that answer is to send
ssh to the background while you call
scp. This may require re-authentication though, but if you setup passkey authentication, this shouldn't be a problem. To send
ssh to the background, hit Ctrl + Z. You should then be able to call
scp, and get
ssh back when you're done, using
Edit: as pointed out by muru, in order to handle Ctrl + Z locally, and not remotely, you'll have to use Enter, ~, Ctrl + Z. See about SSH escape sequences.
ssh just don't work the same way. You'll need a trick to make one do the other's job. Some of those tricks are given in the question I linked earlier.