The way I would do what you're asking for would be to turn on session logging on your ssh client (Ways to do this vary. Depending on the file you might get away with just setting a big scrollback buffer.), and then base64 encode the file and blit it to your terminal. :D
You can then trim the extraneous bits off of either end of the log and base64 decode it back into your original file.
base64 < file.to.transfer
And then, on the client end:
base64 -d < trimmed.log.file > destination.file
Some client programs have methods built in for doing pretty much the same thing with, say, the XMODEM protocol or similar, but availability of the transmission software on the server side is probably going to be a lot less than for base64, which is included in most of the Linux distros I've used recently. If you have a client that supports it though, it takes care of the chopping and decoding phase automatically, which can be convenient for large files. I don't know that anybody's really used it since the days of telnet BBSs though, so good luck.