Slow draining Clothes Washer Drain
Some washing machines have strong pumps that discharge lots of water quickly, overcoming the capacity of the drain standpipe. Yes, unless you can see it on the outside of the wall, there is a standpipe in the wall. You should consider increasing the diameter of the standpipe, thereby increasing the volume of water it can hold as the discharged water drains. I frequently use plastic pipe (PVC, ABS) and fittings for this. 

This is not to say you don't need to also clear the drain, to do this you need a good quality 1/4" drum auger with a stiff cable. The good ones are expensive but will last a lifetime if used and maintained properly. I have had the same one since 1996 and have saved hundreds (thousands?) of $$ by clearing virtually every local clog myself (my neighbors' too).

There is some skill involved. Go to an actual plumbing shop to find a high quality auger and to get good advice on how to use it properly (or buy one and post another question about how to use it).

The good ones look like this:


Not like this: