Up sh*t creek...

You said you only had a 20' snake... Get a longer snake... a power snake with a cutter head would be best.

sounds like it`s out in the yard, go rent a sewer machine, be careful don`t feed a lot of cable where it can flip around and grab u, wear heavy leather gloves--no cotton or such gloves---
like I said before get in to the main line coming out of ur house regardless of what it comes out from under, cut a hole in it big enough for the snake to go thru, go both ways till u hit the clog, then patch the hole. as some stated before a double clean-out tee would be the best, but could be a lot of trouble for some one inexperianced w/ that sort of thing---good luck