Do I have to push the snake as far as possible or can I consider it done if I collected guck and drain now flowing smoothly?

If when you cycle the washer with no problems then you've cleared the blockage. Good job! Cheers, Tom

Yep, Toms correct. As long as its draining fine, your are good to go. It won't hurt a thing to snake further, but no need if blockage is cleared.